Daily Papers

We show that learning-rate schedules for large model training behave surprisingly similar to a performance bound from non-smooth convex optimization theory. We provide a bound for the constant schedule with linear cooldown; in particular, the practical benefit of cooldown is reflected in the bound due to the absence of logarithmic terms. Further, we show that this surprisingly close match between optimization theory and practice can be exploited for learning-rate tuning: we achieve noticeable improvements for training 124M and 210M Llama-type models by (i) extending the schedule for continued training with optimal learning-rate, and (ii) transferring the optimal learning-rate across schedules.

Learning rate schedules used in practice bear little resemblance to those recommended by theory. We close much of this theory/practice gap, and as a consequence are able to derive new problem-adaptive learning rate schedules. Our key technical contribution is a refined analysis of learning rate schedules for a wide class of optimization algorithms (including SGD). In contrast to most prior works that study the convergence of the average iterate, we study the last iterate, which is what most people use in practice. When considering only worst-case analysis, our theory predicts that the best choice is the linear decay schedule: a popular choice in practice that sets the stepsize proportionally to 1 - t/T, where t is the current iteration and T is the total number of steps. To go beyond this worst-case analysis, we use the observed gradient norms to derive schedules refined for any particular task. These refined schedules exhibit learning rate warm-up and rapid learning rate annealing near the end of training. Ours is the first systematic approach to automatically yield both of these properties. We perform the most comprehensive evaluation of learning rate schedules to date, evaluating across 10 diverse deep learning problems, a series of LLMs, and a suite of logistic regression problems. We validate that overall, the linear-decay schedule matches or outperforms all commonly used default schedules including cosine annealing, and that our schedule refinement method gives further improvements.

Learning rate scheduling is essential in transformer training, where the final annealing plays a crucial role in getting the best performance. However, the mechanisms behind this cooldown phase, with its characteristic drop in loss, remain poorly understood. To address this, we provide a comprehensive analysis focusing solely on the cooldown phase in the Warmup-Stable-Decay (WSD) learning rate scheduler. Our analysis reveals that different cooldown shapes reveal a fundamental bias-variance trade-off in the resulting models, with shapes that balance exploration and exploitation consistently outperforming alternatives. Similarly, we find substantial performance variations x2013 comparable to those from cooldown shape selection x2013 when tuning AdamW hyperparameters. Notably, we observe consistent improvements with higher values of beta_2 during cooldown. From a loss landscape perspective, we provide visualizations of the landscape during cooldown, supporting the river valley loss perspective empirically. These findings offer practical recommendations for configuring the WSD scheduler in transformer training, emphasizing the importance of optimizing the cooldown phase alongside traditional hyperparameter tuning.

Learning Rate Rewinding (LRR) has been established as a strong variant of Iterative Magnitude Pruning (IMP) to find lottery tickets in deep overparameterized neural networks. While both iterative pruning schemes couple structure and parameter learning, understanding how LRR excels in both aspects can bring us closer to the design of more flexible deep learning algorithms that can optimize diverse sets of sparse architectures. To this end, we conduct experiments that disentangle the effect of mask learning and parameter optimization and how both benefit from overparameterization. The ability of LRR to flip parameter signs early and stay robust to sign perturbations seems to make it not only more effective in mask identification but also in optimizing diverse sets of masks, including random ones. In support of this hypothesis, we prove in a simplified single hidden neuron setting that LRR succeeds in more cases than IMP, as it can escape initially problematic sign configurations.

We design learning rate schedules that minimize regret for SGD-based online learning in the presence of a changing data distribution. We fully characterize the optimal learning rate schedule for online linear regression via a novel analysis with stochastic differential equations. For general convex loss functions, we propose new learning rate schedules that are robust to distribution shift, and we give upper and lower bounds for the regret that only differ by constants. For non-convex loss functions, we define a notion of regret based on the gradient norm of the estimated models and propose a learning schedule that minimizes an upper bound on the total expected regret. Intuitively, one expects changing loss landscapes to require more exploration, and we confirm that optimal learning rate schedules typically increase in the presence of distribution shift. Finally, we provide experiments for high-dimensional regression models and neural networks to illustrate these learning rate schedules and their cumulative regret.

D-Adaptation is an approach to automatically setting the learning rate which asymptotically achieves the optimal rate of convergence for minimizing convex Lipschitz functions, with no back-tracking or line searches, and no additional function value or gradient evaluations per step. Our approach is the first hyper-parameter free method for this class without additional multiplicative log factors in the convergence rate. We present extensive experiments for SGD and Adam variants of our method, where the method automatically matches hand-tuned learning rates across more than a dozen diverse machine learning problems, including large-scale vision and language problems. An open-source implementation is available.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

In current deep learning tasks, Adam style optimizers such as Adam, Adagrad, RMSProp, Adafactor, and Lion have been widely used as alternatives to SGD style optimizers. These optimizers typically update model parameters using the sign of gradients, resulting in more stable convergence curves. The learning rate and the batch size are the most critical hyperparameters for optimizers, which require careful tuning to enable effective convergence. Previous research has shown that the optimal learning rate increases linearly or follows similar rules with batch size for SGD style optimizers. However, this conclusion is not applicable to Adam style optimizers. In this paper, we elucidate the connection between optimal learning rates and batch sizes for Adam style optimizers through both theoretical analysis and extensive experiments. First, we raise the scaling law between batch sizes and optimal learning rates in the sign of gradient case, in which we prove that the optimal learning rate first rises and then falls as the batch size increases. Moreover, the peak value of the surge will gradually move toward the larger batch size as training progresses. Second, we conducted experiments on various CV and NLP tasks and verified the correctness of the scaling law.

It is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically eliminates the need to experimentally find the best values and schedule for the global learning rates. Instead of monotonically decreasing the learning rate, this method lets the learning rate cyclically vary between reasonable boundary values. Training with cyclical learning rates instead of fixed values achieves improved classification accuracy without a need to tune and often in fewer iterations. This paper also describes a simple way to estimate "reasonable bounds" -- linearly increasing the learning rate of the network for a few epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10 and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets, and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These are practical tools for everyone who trains neural networks.

Effective learning rate (LR) scheduling is crucial for training deep neural networks. However, popular pre-defined and adaptive schedulers can still lead to suboptimal generalization. This paper introduces VolSched, a novel adaptive LR scheduler inspired by the concept of volatility in stochastic processes like Geometric Brownian Motion to dynamically adjust the learning rate. By calculating the ratio between long-term and short-term accuracy volatility, VolSched increases the LR to escape plateaus and decreases it to stabilize training, allowing the model to explore the loss landscape more effectively. We evaluate VolSched on the CIFAR-100 dataset against a strong baseline using a standard augmentation pipeline. When paired with ResNet-18 and ResNet-34, our scheduler delivers consistent performance gains, improving top-1 accuracy by 1.4 and 1.3 percentage points respectively. Analysis of the loss curves reveals that VolSched promotes a longer exploration phase. A quantitative analysis of the Hessian shows that VolSched finds a final solution that is 38% flatter than the next-best baseline, allowing the model to obtain wider minima and hence better generalization performance.

In this work, we theoretically investigate the generalization properties of neural networks (NN) trained by stochastic gradient descent (SGD) algorithm with large learning rates. Under such a training regime, our finding is that, the oscillation of the NN weights caused by the large learning rate SGD training turns out to be beneficial to the generalization of the NN, which potentially improves over the same NN trained by SGD with small learning rates that converges more smoothly. In view of this finding, we call such a phenomenon "benign oscillation". Our theory towards demystifying such a phenomenon builds upon the feature learning perspective of deep learning. Specifically, we consider a feature-noise data generation model that consists of (i) weak features which have a small ell_2-norm and appear in each data point; (ii) strong features which have a larger ell_2-norm but only appear in a certain fraction of all data points; and (iii) noise. We prove that NNs trained by oscillating SGD with a large learning rate can effectively learn the weak features in the presence of those strong features. In contrast, NNs trained by SGD with a small learning rate can only learn the strong features but makes little progress in learning the weak features. Consequently, when it comes to the new testing data which consist of only weak features, the NN trained by oscillating SGD with a large learning rate could still make correct predictions consistently, while the NN trained by small learning rate SGD fails. Our theory sheds light on how large learning rate training benefits the generalization of NNs. Experimental results demonstrate our finding on "benign oscillation".

Large Language Models (LLMs) represent the recent success of deep learning in achieving remarkable human-like predictive performance. It has become a mainstream strategy to leverage fine-tuning to adapt LLMs for various real-world applications due to the prohibitive expenses associated with LLM training. The learning rate is one of the most important hyperparameters in LLM fine-tuning with direct impacts on both fine-tuning efficiency and fine-tuned LLM quality. Existing learning rate policies are primarily designed for training traditional deep neural networks (DNNs), which may not work well for LLM fine-tuning. We reassess the research challenges and opportunities of learning rate tuning in the coming era of Large Language Models. This paper makes three original contributions. First, we revisit existing learning rate policies to analyze the critical challenges of learning rate tuning in the era of LLMs. Second, we present LRBench++ to benchmark learning rate policies and facilitate learning rate tuning for both traditional DNNs and LLMs. Third, our experimental analysis with LRBench++ demonstrates the key differences between LLM fine-tuning and traditional DNN training and validates our analysis.

The learning rate (LR) schedule is one of the most important hyper-parameters needing careful tuning in training DNNs. However, it is also one of the least automated parts of machine learning systems and usually costs significant manual effort and computing. Though there are pre-defined LR schedules and optimizers with adaptive LR, they introduce new hyperparameters that need to be tuned separately for different tasks/datasets. In this paper, we consider the question: Can we automatically tune the LR over the course of training without human involvement? We propose an efficient method, AutoLRS, which automatically optimizes the LR for each training stage by modeling training dynamics. AutoLRS aims to find an LR applied to every tau steps that minimizes the resulted validation loss. We solve this black-box optimization on the fly by Bayesian optimization (BO). However, collecting training instances for BO requires a system to evaluate each LR queried by BO's acquisition function for tau steps, which is prohibitively expensive in practice. Instead, we apply each candidate LR for only tau'lltau steps and train an exponential model to predict the validation loss after tau steps. This mutual-training process between BO and the loss-prediction model allows us to limit the training steps invested in the BO search. We demonstrate the advantages and the generality of AutoLRS through extensive experiments of training DNNs for tasks from diverse domains using different optimizers. The LR schedules auto-generated by AutoLRS lead to a speedup of 1.22times, 1.43times, and 1.5times when training ResNet-50, Transformer, and BERT, respectively, compared to the LR schedules in their original papers, and an average speedup of 1.31times over state-of-the-art heavily-tuned LR schedules.

The learning rate warmup heuristic achieves remarkable success in stabilizing training, accelerating convergence and improving generalization for adaptive stochastic optimization algorithms like RMSprop and Adam. Here, we study its mechanism in details. Pursuing the theory behind warmup, we identify a problem of the adaptive learning rate (i.e., it has problematically large variance in the early stage), suggest warmup works as a variance reduction technique, and provide both empirical and theoretical evidence to verify our hypothesis. We further propose RAdam, a new variant of Adam, by introducing a term to rectify the variance of the adaptive learning rate. Extensive experimental results on image classification, language modeling, and neural machine translation verify our intuition and demonstrate the effectiveness and robustness of our proposed method. All implementations are available at: https://github.com/LiyuanLucasLiu/RAdam.

We propose Adam-mini, an optimizer that achieves on-par or better performance than AdamW with 45% to 50% less memory footprint. Adam-mini reduces memory by cutting down the learning rate resources in Adam (i.e., 1/v). We find that geq 90% of these learning rates in v could be harmlessly removed if we (1) carefully partition the parameters into blocks following our proposed principle on Hessian structure; (2) assign a single but good learning rate to each parameter block. We further find that, for each of these parameter blocks, there exists a single high-quality learning rate that can outperform Adam, provided that sufficient resources are available to search it out. We then provide one cost-effective way to find good learning rates and propose Adam-mini. Empirically, we verify that Adam-mini performs on par or better than AdamW on various language models sized from 125M to 7B for pre-training, supervised fine-tuning, and RLHF. The reduced memory footprint of Adam-mini also alleviates communication overheads among GPUs and CPUs, thereby increasing throughput. For instance, Adam-mini achieves 49.6% higher throughput than AdamW when pre-training Llama2-7B on 2times A800-80GB GPUs, which saves 33% wall-clock time for pre-training.

In this work, we question the necessity of adaptive gradient methods for training deep neural networks. SGD-SaI is a simple yet effective enhancement to stochastic gradient descent with momentum (SGDM). SGD-SaI performs learning rate Scaling at Initialization (SaI) to distinct parameter groups, guided by their respective gradient signal-to-noise ratios (g-SNR). By adjusting learning rates without relying on adaptive second-order momentum, SGD-SaI helps prevent training imbalances from the very first iteration and cuts the optimizer's memory usage by half compared to AdamW. Despite its simplicity and efficiency, SGD-SaI consistently matches or outperforms AdamW in training a variety of Transformer-based tasks, effectively overcoming a long-standing challenge of using SGD for training Transformers. SGD-SaI excels in ImageNet-1K classification with Vision Transformers(ViT) and GPT-2 pretraining for large language models (LLMs, transformer decoder-only), demonstrating robustness to hyperparameter variations and practicality for diverse applications. We further tested its robustness on tasks like LoRA fine-tuning for LLMs and diffusion models, where it consistently outperforms state-of-the-art optimizers. From a memory efficiency perspective, SGD-SaI achieves substantial memory savings for optimizer states, reducing memory usage by 5.93 GB for GPT-2 (1.5B parameters) and 25.15 GB for Llama2-7B compared to AdamW in full-precision training settings.

Training large language models (LLMs) poses challenges due to their massive scale and heterogeneous architectures. While adaptive optimizers like AdamW help address gradient variations, they still struggle with efficient and effective parameter-wise learning rate estimation, resulting in training instability, slow convergence, and poor compatibility with parameter-efficient fine-tuning (PEFT) techniques. This work introduces Scaling with Gradient Grouping (SGG), an optimizer wrapper that improves adaptive learning rate estimation by dynamic grouping and group-specific scaling. SGG first groups gradient statistics in each layer into clusters and then applies cluster-specific scaling to calibrate learning rates for each parameter, thus imposing collective group-wise constraints while maintaining precise per-parameter adaptation. Experiments on diverse (M)LLM benchmarks show that SGG integrates seamlessly with existing optimizers, and offers consistent gains and faster convergence over baselines, with various model sizes. Its stability across varying batch sizes and learning rates establishes SGG as a robust choice for LLM optimization.

Finding the optimal learning rate for language model pretraining is a challenging task. This is not only because there is a complicated correlation between learning rate, batch size, number of training tokens, model size, and other hyperparameters but also because it is prohibitively expensive to perform a hyperparameter search for large language models with Billions or Trillions of parameters. Recent studies propose using small proxy models and small corpus to perform hyperparameter searches and transposing the optimal parameters to large models and large corpus. While the zero-shot transferability is theoretically and empirically proven for model size related hyperparameters, like depth and width, the zero-shot transfer from small corpus to large corpus is underexplored. In this paper, we study the correlation between optimal learning rate, batch size, and number of training tokens for the recently proposed WSD scheduler. After thousands of small experiments, we found a power-law relationship between variables and demonstrated its transferability across model sizes. Based on the observation, we propose a new learning rate scheduler, Power scheduler, that is agnostic about the number of training tokens and batch size. The experiment shows that combining the Power scheduler with Maximum Update Parameterization (muP) can consistently achieve impressive performance with one set of hyperparameters regardless of the number of training tokens, batch size, model size, and even model architecture. Our 3B dense and MoE models trained with the Power scheduler achieve comparable performance as state-of-the-art small language models. We open-source these pretrained models at https://ibm.biz/BdKhLa.

The expanding computational costs and limited resources underscore the critical need for budgeted-iteration training, which aims to achieve optimal learning within predetermined iteration budgets.While learning rate schedules fundamentally govern the performance of different networks and tasks, particularly in budgeted-iteration scenarios, their design remains largely heuristic, lacking theoretical foundations.In addition, the optimal learning rate schedule requires extensive trial-and-error selection, making the training process inefficient.In this work, we propose the Unified Budget-Aware (UBA) schedule, a theoretically grounded learning rate schedule that consistently outperforms commonly-used schedules among diverse architectures and tasks under different constrained training budgets.First, we bridge the gap by constructing a novel training budget-aware optimization framework, which explicitly accounts for the robustness to landscape curvature variations.From this framework, we derive the UBA schedule, controlled by a single hyper-parameter varphi that provides a trade-off between flexibility and simplicity, eliminating the need for per-network numerical optimization. Moreover, we establish a theoretical connection between varphi and the condition number, adding interpretation and justification to our approach. Besides, we prove the convergence for different values of varphi.We offer practical guidelines for its selection via theoretical analysis and empirical results.xtensive experimental results show that UBA consistently surpasses the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets.

We present SmolTulu-1.7b-Instruct, referenced in this report as SmolTulu-DPO-1130, an instruction-tuned language model that adapts AllenAI's Tulu 3 post-training pipeline to enhance Huggingface's SmolLM2-1.7B base model. Through comprehensive empirical analysis using a 135M parameter model, we demonstrate that the relationship between learning rate and batch size significantly impacts model performance in a task-dependent manner. Our findings reveal a clear split: reasoning tasks like ARC and GSM8K benefit from higher learning rate to batch size ratios, while pattern recognition tasks such as HellaSwag and IFEval show optimal performance with lower ratios. These insights informed the development of SmolTulu, which achieves state-of-the-art performance among sub-2B parameter models on instruction following, scoring 67.7% on IFEval (Delta11%), and mathematical reasoning with 51.6% on GSM8K (Delta3.4%), with an alternate version achieving scoring 57.1% on ARC (Delta5.4%). We release our model, training recipes, and ablation studies to facilitate further research in efficient model alignment, demonstrating that careful adaptation of optimization dynamics can help bridge the capability gap between small and large language models.

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps (s): $L(s) = L_0 + Acdot S_1^{-alpha} - Ccdot S_2 Where S_1 is forward area and S_2$ is learning rate annealing area. This formulation takes into account two factors: (1) The forward scaling defined as typical scaling law, and (2) the additional loss drop brought by LR annealing. Therefore, this formulation can describe the full loss curve at each step, rather than the single loss point at the end of training. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss of language model training at any given step and across any learning rate scheduler (LRS). Furthermore, this equation accurately describes the dynamics during training process, and provides a theoretical verification and explanation for numerous experimental findings of previous studies, particularly those focusing on LR schedule and LR annealing. The resulting insights, also serve as a guide for researchers to select critical LRS in advance by prediction using our equation. Most significantly, since all the points in a full training curve follow the equation, we can achieve accurate loss prediction at any given step across any learning rate scheduler, while expending less than 1\% of the computational cost required by the chinchilla scaling law to fit language modeling loss. This approach extremely democratizes scaling law fitting and predicting in developing large language models.

Recent advances in learning rate (LR) scheduling have demonstrated the effectiveness of decay-free approaches that eliminate the traditional decay phase while maintaining competitive performance. Model merging techniques have emerged as particularly promising solutions in this domain. We present Warmup-Stable and Merge (WSM), a general framework that establishes a formal connection between learning rate decay and model merging. WSM provides a unified theoretical foundation for emulating various decay strategies-including cosine decay, linear decay and inverse square root decay-as principled model averaging schemes, while remaining fully compatible with diverse optimization methods. Through extensive experiments, we identify merge duration-the training window for checkpoint aggregation-as the most critical factor influencing model performance, surpassing the importance of both checkpoint interval and merge quantity. Our framework consistently outperforms the widely-adopted Warmup-Stable-Decay (WSD) approach across multiple benchmarks, achieving significant improvements of +3.5% on MATH, +2.9% on HumanEval, and +5.5% on MMLU-Pro. The performance advantages extend to supervised fine-tuning scenarios, highlighting WSM's potential for long-term model refinement.

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

The ever-growing availability of unlabeled data presents both opportunities and challenges for training artificial intelligence systems. While self-supervised learning (SSL) has emerged as a powerful paradigm for extracting meaningful representations from vast amounts of unlabeled data, existing methods still struggle to adapt to the non-stationary, non-IID nature of real-world data streams without forgetting previously learned knowledge. Recent works have adopted a repeated cosine annealing schedule for large-scale continual pre-training; however, these schedules (1) inherently cause forgetting during the re-warming phase and (2) have not been systematically compared to existing continual SSL methods. In this work, we systematically compare the widely used cosine schedule with the recently proposed infinite learning rate schedule and empirically find the latter to be a more effective alternative. Our extensive empirical evaluation across diverse image and language datasets demonstrates that the infinite learning rate schedule consistently enhances continual pre-training performance compared to a repeated cosine decay without being restricted to a fixed iteration budget. For instance, in a small-scale MAE pre-training setup, it outperforms several strong baselines from the literature. We then scale up our experiments to larger MAE pre-training and autoregressive language model pre-training. Our results show that the infinite learning rate schedule remains effective at scale, surpassing repeated cosine decay for both MAE pre-training and zero-shot LM benchmarks.

One of the main challenges in optimal scaling of large language models (LLMs) is the prohibitive cost of hyperparameter tuning, particularly learning rate eta and batch size B. While techniques like muP (Yang et al., 2022) provide scaling rules for optimal eta transfer in the infinite model size limit, the optimal scaling behavior in the infinite data size limit remains unknown. We fill in this gap by observing for the first time an intricate dependence of optimal eta scaling on the pretraining token budget T, B and its relation to the critical batch size B_crit, which we measure to evolve as B_crit propto T. Furthermore, we show that the optimal batch size is positively correlated with B_crit: keeping it fixed becomes suboptimal over time even if learning rate is scaled optimally. Surprisingly, our results demonstrate that the observed optimal eta and B dynamics are preserved with muP model scaling, challenging the conventional view of B_crit dependence solely on loss value. Complementing optimality, we examine the sensitivity of loss to changes in learning rate, where we find the sensitivity to decrease with increase of T and to remain constant with muP model scaling. We hope our results make the first step towards a unified picture of the joint optimal data and model scaling.

Training language models currently requires pre-determining a fixed compute budget because the typical cosine learning rate schedule depends on the total number of steps. In contrast, the Warmup-Stable-Decay (WSD) schedule uses a constant learning rate to produce a main branch of iterates that can in principle continue indefinitely without a pre-specified compute budget. Then, given any compute budget, one can branch out from the main branch at a proper time with a rapidly decaying learning rate to produce a strong model. Empirically, WSD generates a non-traditional loss curve: the loss remains elevated during the stable phase but sharply declines during the decay phase. Towards explaining this phenomenon, we conjecture that pretraining loss exhibits a river valley landscape, which resembles a deep valley with a river at its bottom. Under this assumption, we show that during the stable phase, the iterate undergoes large oscillations due to the high learning rate, yet it progresses swiftly along the river. During the decay phase, the rapidly dropping learning rate minimizes the iterate's oscillations, moving it closer to the river and revealing true optimization progress. Therefore, the sustained high learning rate phase and fast decaying phase are responsible for progress in the river and the mountain directions respectively, and are both critical. Our analysis predicts phenomenons consistent with empirical observations and shows that this landscape can emerge from pretraining on a simple bi-gram dataset. Inspired by the theory, we introduce WSD-S, a variant of WSD that reuses previous checkpoints' decay phases and keeps only one main branch, where we resume from a decayed checkpoint. WSD-S empirically outperforms WSD and Cyclic-Cosine in obtaining multiple language model checkpoints across various compute budgets in a single run for parameters scaling from 0.1B to 1.2B.

Low-Rank Adaptation (LoRA) is the bread and butter of Large Language Model (LLM) finetuning. LoRA learns an additive low-rank perturbation, AB, of a pretrained matrix parameter W to align the model to a new task or dataset with W+AB. We identify three core limitations to LoRA for finetuning--a setting that employs limited amount of data and training steps. First, LoRA employs Dropout to prevent overfitting. We prove that Dropout is only suitable for long training episodes but fails to converge to a reliable regularizer for short training episodes. Second, LoRA's initialization of B at 0 creates a slow training dynamic between A and B. That dynamic is also exacerbated by Dropout that further slows the escape from 0 for B which is particularly harmful for short training episodes. Third, the scaling factor multiplying each LoRA additive perturbation creates ``short-sighted'' interactions between the LoRA modules of different layers. Motivated by principled analysis of those limitations, we find an elegant solution: a Dropout-free, scaling-free, LoRA with Adaptive Learning rate--coined ALLoRA. By scaling the per sample and per parameter gradients with a coefficient inversely proportional to parameters' ell_2 norm, ALLoRA alleviates those three limitations. As a by-product, ALLoRA removes two hyper-parameters from LoRA: the scaling factor and the dropout rate. Empirical results show that ALLoRA admits better accuracy than LoRA on various settings, including against recent LoRA variants such as Weight-Decomposed Low-Rank Adaptation (DoRA). Ablation studies show our solution is the optimal in a family of weight-dependent / output-dependent approaches on various LLMs including the latest Llama3.

Normalization layers have recently experienced a renaissance in the deep reinforcement learning and continual learning literature, with several works highlighting diverse benefits such as improving loss landscape conditioning and combatting overestimation bias. However, normalization brings with it a subtle but important side effect: an equivalence between growth in the norm of the network parameters and decay in the effective learning rate. This becomes problematic in continual learning settings, where the resulting effective learning rate schedule may decay to near zero too quickly relative to the timescale of the learning problem. We propose to make the learning rate schedule explicit with a simple re-parameterization which we call Normalize-and-Project (NaP), which couples the insertion of normalization layers with weight projection, ensuring that the effective learning rate remains constant throughout training. This technique reveals itself as a powerful analytical tool to better understand learning rate schedules in deep reinforcement learning, and as a means of improving robustness to nonstationarity in synthetic plasticity loss benchmarks along with both the single-task and sequential variants of the Arcade Learning Environment. We also show that our approach can be easily applied to popular architectures such as ResNets and transformers while recovering and in some cases even slightly improving the performance of the base model in common stationary benchmarks.

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit (muP and its depth extension), then some hyperparameters - such as the learning rate - exhibit transfer from small to very large models, thus reducing the cost of hyperparameter tuning. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is remarkably consistent across very different model sizes. In this work, we find empirical evidence that learning rate transfer can be attributed to the fact that under muP and its depth extension, the largest eigenvalue of the training loss Hessian (i.e. the sharpness) is largely independent of the width and depth of the network for a sustained period of training time. On the other hand, we show that under the neural tangent kernel (NTK) regime, the sharpness exhibits very different dynamics at different scales, thus preventing learning rate transfer. But what causes these differences in the sharpness dynamics? Through a connection between the spectra of the Hessian and the NTK matrix, we argue that the cause lies in the presence (for muP) or progressive absence (for the NTK regime) of feature learning, which results in a different evolution of the NTK, and thus of the sharpness. We corroborate our claims with a substantial suite of experiments, covering a wide range of datasets and architectures: from ResNets and Vision Transformers trained on benchmark vision datasets to Transformers-based language models trained on WikiText

Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate eta^{ast} - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate eta^{ast}, we observe that in constant-width fully-connected ReLU networks, eta^{ast} behaves differently from the maximum learning rate later in training. Specifically, we find that eta^{ast} is well predicted as a power of depth times width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between eta^{ast} and the sharpness lambda_{1} of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for lambda_{1} in terms of depth times width that align with our empirical results.

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is multiplicative, we show that the learners can, in fact, achieve constant regret. We achieve this faster rate via an optimistic gradient scheme with learning rate separation -- that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that attains nearly the same guarantees as its non-adapted counterpart, while operating without knowledge of either the game or of the noise profile.

Benchmarking the tradeoff between neural network accuracy and training time is computationally expensive. Here we show how a multiplicative cyclic learning rate schedule can be used to construct a tradeoff curve in a single training run. We generate cyclic tradeoff curves for combinations of training methods such as Blurpool, Channels Last, Label Smoothing and MixUp, and highlight how these cyclic tradeoff curves can be used to evaluate the effects of algorithmic choices on network training efficiency.

Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new adaptive learning rates that can be used with any momentum method, and require less tuning to perform well. We first develop MoMo, a Momentum Model based adaptive learning rate for SGD-M (Stochastic gradient descent with momentum). MoMo uses momentum estimates of the batch losses and gradients sampled at each iteration to build a model of the loss function. Our model also makes use of any known lower bound of the loss function by using truncation, e.g. most losses are lower-bounded by zero. We then approximately minimize this model at each iteration to compute the next step. We show how MoMo can be used in combination with any momentum-based method, and showcase this by developing MoMo-Adam - which is Adam with our new model-based adaptive learning rate. Additionally, for losses with unknown lower bounds, we develop on-the-fly estimates of a lower bound, that are incorporated in our model. Through extensive numerical experiments, we demonstrate that MoMo and MoMo-Adam improve over SGD-M and Adam in terms of accuracy and robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR10, CIFAR100, Imagenet, recommender systems on the Criteo dataset, and a transformer model on the translation task IWSLT14.

Large language models have achieved remarkable success, but their extensive parameter size necessitates substantial memory for training, thereby setting a high threshold. While the recently proposed low-memory optimization (LOMO) reduces memory footprint, its optimization technique, akin to stochastic gradient descent, is sensitive to hyper-parameters and exhibits suboptimal convergence, failing to match the performance of the prevailing optimizer for large language models, AdamW. Through empirical analysis of the Adam optimizer, we found that, compared to momentum, the adaptive learning rate is more critical for bridging the gap. Building on this insight, we introduce the low-memory optimization with adaptive learning rate (AdaLomo), which offers an adaptive learning rate for each parameter. To maintain memory efficiency, we employ non-negative matrix factorization for the second-order moment estimation in the optimizer state. Additionally, we suggest the use of a grouped update normalization to stabilize convergence. Our experiments with instruction-tuning and further pre-training demonstrate that AdaLomo achieves results on par with AdamW, while significantly reducing memory requirements, thereby lowering the hardware barrier to training large language models.

Although deep learning has produced dazzling successes for applications of image, speech, and video processing in the past few years, most trainings are with suboptimal hyper-parameters, requiring unnecessarily long training times. Setting the hyper-parameters remains a black art that requires years of experience to acquire. This report proposes several efficient ways to set the hyper-parameters that significantly reduce training time and improves performance. Specifically, this report shows how to examine the training validation/test loss function for subtle clues of underfitting and overfitting and suggests guidelines for moving toward the optimal balance point. Then it discusses how to increase/decrease the learning rate/momentum to speed up training. Our experiments show that it is crucial to balance every manner of regularization for each dataset and architecture. Weight decay is used as a sample regularizer to show how its optimal value is tightly coupled with the learning rates and momentums. Files to help replicate the results reported here are available.

LLMs are commonly trained with a learning rate (LR) warmup, followed by cosine decay to 10% of the maximum (10x decay). In a large-scale empirical study, we show that under an optimal peak LR, a simple linear decay-to-zero (D2Z) schedule consistently outperforms other schedules when training at compute-optimal dataset sizes. D2Z is superior across a range of model sizes, batch sizes, datasets, and vocabularies. Benefits increase as dataset size increases. Leveraging a novel interpretation of AdamW as an exponential moving average of weight updates, we show how linear D2Z optimally balances the demands of early training (moving away from initial conditions) and late training (averaging over more updates in order to mitigate gradient noise). In experiments, a 610M-parameter model trained for 80 tokens-per-parameter (TPP) using D2Z achieves lower loss than when trained for 200 TPP using 10x decay, corresponding to an astonishing 60% compute savings. Models such as Llama2-7B, trained for 286 TPP with 10x decay, could likely have saved a majority of compute by training with D2Z.

The theoretical landscape of federated learning (FL) undergoes rapid evolution, but its practical application encounters a series of intricate challenges, and hyperparameter optimization is one of these critical challenges. Amongst the diverse adjustments in hyperparameters, the adaptation of the learning rate emerges as a crucial component, holding the promise of significantly enhancing the efficacy of FL systems. In response to this critical need, this paper presents FedHyper, a novel hypergradient-based learning rate adaptation algorithm specifically designed for FL. FedHyper serves as a universal learning rate scheduler that can adapt both global and local rates as the training progresses. In addition, FedHyper not only showcases unparalleled robustness to a spectrum of initial learning rate configurations but also significantly alleviates the necessity for laborious empirical learning rate adjustments. We provide a comprehensive theoretical analysis of FedHyper's convergence rate and conduct extensive experiments on vision and language benchmark datasets. The results demonstrate that FEDHYPER consistently converges 1.1-3x faster than FedAvg and the competing baselines while achieving superior final accuracy. Moreover, FedHyper catalyzes a remarkable surge in accuracy, augmenting it by up to 15% compared to FedAvg under suboptimal initial learning rate settings.

Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.

When fitting the learning data of an individual to algorithm-like learning models, the observations are so dependent and non-stationary that one may wonder what the classical Maximum Likelihood Estimator (MLE) could do, even if it is the usual tool applied to experimental cognition. Our objective in this work is to show that the estimation of the learning rate cannot be efficient if the learning rate is constant in the classical Exp3 (Exponential weights for Exploration and Exploitation) algorithm. Secondly, we show that if the learning rate decreases polynomially with the sample size, then the prediction error and in some cases the estimation error of the MLE satisfy bounds in probability that decrease at a polynomial rate.

In recent years, particle-based variational inference (ParVI) methods such as Stein variational gradient descent (SVGD) have grown in popularity as scalable methods for Bayesian inference. Unfortunately, the properties of such methods invariably depend on hyperparameters such as the learning rate, which must be carefully tuned by the practitioner in order to ensure convergence to the target measure at a suitable rate. In this paper, we introduce a suite of new particle-based methods for scalable Bayesian inference based on coin betting, which are entirely learning-rate free. We illustrate the performance of our approach on a range of numerical examples, including several high-dimensional models and datasets, demonstrating comparable performance to other ParVI algorithms with no need to tune a learning rate.

When training neural networks, it has been widely observed that a large step size is essential in stochastic gradient descent (SGD) for obtaining superior models. However, the effect of large step sizes on the success of SGD is not well understood theoretically. Several previous works have attributed this success to the stochastic noise present in SGD. However, we show through a novel set of experiments that the stochastic noise is not sufficient to explain good non-convex training, and that instead the effect of a large learning rate itself is essential for obtaining best performance.We demonstrate the same effects also in the noise-less case, i.e. for full-batch GD. We formally prove that GD with large step size -- on certain non-convex function classes -- follows a different trajectory than GD with a small step size, which can lead to convergence to a global minimum instead of a local one. Our settings provide a framework for future analysis which allows comparing algorithms based on behaviors that can not be observed in the traditional settings.

Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSGrad, called AdaBound and AMSBound respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound .

Training Large Language Models (LLMs) incurs significant cost; hence, any strategy that accelerates model convergence is helpful. In this paper, we investigate the ability of a simple idea checkpoint averaging along the trajectory of a training run to improve both convergence and generalization quite early on during training. Here we show that models trained with high learning rates observe higher gains due to checkpoint averaging. Furthermore, these gains are amplified when checkpoints are sampled with considerable spacing in training steps. Our training recipe outperforms conventional training and popular checkpoint averaging baselines such as exponential moving average (EMA) and stochastic moving average (SWA). We evaluate our training recipe by pre-training LLMs, where high learning rates are inherently preferred due to extremely large batch sizes. Specifically, we pre-trained nanoGPT-2 models of varying sizes, small (125M), medium (335M), and large (770M)on the OpenWebText dataset, comprised of 9B tokens. Additionally, we present results for publicly available Pythia LLMs, ranging from 1B to 12B, which were trained on the PILE-deduped dataset containing 207B tokens.

Training large models is both resource-intensive and time-consuming, making it crucial to understand the quantitative relationship between model performance and hyperparameters. In this paper, we present an empirical law that describes how the pretraining loss of large language models evolves under different learning rate schedules, such as constant, cosine, and step decay schedules. Our proposed law takes a multi-power form, combining a power law based on the sum of learning rates and additional power laws to account for a loss reduction effect induced by learning rate decay. We extensively validate this law on various model sizes and architectures, and demonstrate that after fitting on a few learning rate schedules, the law accurately predicts the loss curves for unseen schedules of different shapes and horizons. Moreover, by minimizing the predicted final pretraining loss across learning rate schedules, we are able to find a schedule that outperforms the widely used cosine learning rate schedule. Interestingly, this automatically discovered schedule bears some resemblance to the recently proposed Warmup-Stable-Decay (WSD) schedule (Hu et al, 2024) but achieves a slightly lower final loss. We believe these results could offer valuable insights for understanding the dynamics of pretraining and designing learning rate schedules to improve efficiency.

In this paper, we describe a phenomenon, which we named "super-convergence", where neural networks can be trained an order of magnitude faster than with standard training methods. The existence of super-convergence is relevant to understanding why deep networks generalize well. One of the key elements of super-convergence is training with one learning rate cycle and a large maximum learning rate. A primary insight that allows super-convergence training is that large learning rates regularize the training, hence requiring a reduction of all other forms of regularization in order to preserve an optimal regularization balance. We also derive a simplification of the Hessian Free optimization method to compute an estimate of the optimal learning rate. Experiments demonstrate super-convergence for Cifar-10/100, MNIST and Imagenet datasets, and resnet, wide-resnet, densenet, and inception architectures. In addition, we show that super-convergence provides a greater boost in performance relative to standard training when the amount of labeled training data is limited. The architectures and code to replicate the figures in this paper are available at github.com/lnsmith54/super-convergence. See http://www.fast.ai/2018/04/30/dawnbench-fastai/ for an application of super-convergence to win the DAWNBench challenge (see https://dawn.cs.stanford.edu/benchmark/).

Despite major advances in methodology, hyperparameter tuning remains a crucial (and expensive) part of the development of machine learning systems. Even ignoring architectural choices, deep neural networks have a large number of optimization and regularization hyperparameters that need to be tuned carefully per workload in order to obtain the best results. In a perfect world, training algorithms would not require workload-specific hyperparameter tuning, but would instead have default settings that performed well across many workloads. Recently, there has been a growing literature on optimization methods which attempt to reduce the number of hyperparameters -- particularly the learning rate and its accompanying schedule. Given these developments, how far away is the dream of neural network training algorithms that completely obviate the need for painful tuning? In this paper, we evaluate the potential of learning-rate-free methods as components of hyperparameter-free methods. We freeze their (non-learning rate) hyperparameters to default values, and score their performance using the recently-proposed AlgoPerf: Training Algorithms benchmark. We found that literature-supplied default settings performed poorly on the benchmark, so we performed a search for hyperparameter configurations that performed well across all workloads simultaneously. The best AlgoPerf-calibrated learning-rate-free methods had much improved performance but still lagged slightly behind a similarly calibrated NadamW baseline in overall benchmark score. Our results suggest that there is still much room for improvement for learning-rate-free methods, and that testing against a strong, workload-agnostic baseline is important to improve hyperparameter reduction techniques.

When an agent encounters a continual stream of new tasks in the lifelong learning setting, it leverages the knowledge it gained from the earlier tasks to help learn the new tasks better. In such a scenario, identifying an efficient knowledge representation becomes a challenging problem. Most research works propose to either store a subset of examples from the past tasks in a replay buffer, dedicate a separate set of parameters to each task or penalize excessive updates over parameters by introducing a regularization term. While existing methods employ the general task-agnostic stochastic gradient descent update rule, we propose a task-aware optimizer that adapts the learning rate based on the relatedness among tasks. We utilize the directions taken by the parameters during the updates by accumulating the gradients specific to each task. These task-based accumulated gradients act as a knowledge base that is maintained and updated throughout the stream. We empirically show that our proposed adaptive learning rate not only accounts for catastrophic forgetting but also allows positive backward transfer. We also show that our method performs better than several state-of-the-art methods in lifelong learning on complex datasets with a large number of tasks.

Training deep neural networks requires intricate initialization and careful selection of learning rates. The emergence of stochastic gradient optimization methods that use adaptive learning rates based on squared past gradients, e.g., AdaGrad, AdaDelta, and Adam, eases the job slightly. However, such methods have also been proven problematic in recent studies with their own pitfalls including non-convergence issues and so on. Alternative variants have been proposed for enhancement, such as AMSGrad, AdaShift and AdaBound. In this work, we identify a new problem of adaptive learning rate methods that exhibits at the beginning of learning where Adam produces extremely large learning rates that inhibit the start of learning. We propose the Adaptive and Momental Bound (AdaMod) method to restrict the adaptive learning rates with adaptive and momental upper bounds. The dynamic learning rate bounds are based on the exponential moving averages of the adaptive learning rates themselves, which smooth out unexpected large learning rates and stabilize the training of deep neural networks. Our experiments verify that AdaMod eliminates the extremely large learning rates throughout the training and brings significant improvements especially on complex networks such as DenseNet and Transformer, compared to Adam. Our implementation is available at: https://github.com/lancopku/AdaMod

We present a self-supervised U-Net-based masked autoencoder and noise removal model designed to reconstruct original images. Once adequately trained, this model extracts high-level features, which are then combined with features from the EfficientNet B7 model. These integrated features are subsequently fed into dense layers for classification. Among the approaches of masked input and Gaussian noise removal, we selected the best U-Net reconstruction model. Additionally, we explored various configurations, including EfficientNet with attention, attention fusion of the autoencoder, and classification utilizing U-Net encoder features. The best performance was achieved with EfficientNet B7 combined with U-Net encoder features. We employed the Adam optimizer with a learning rate of 0.0001, achieving a top accuracy of 0.94 on the validation set.

The goal of continual learning is to improve the performance of recognition models in learning sequentially arrived data. Although most existing works are established on the premise of learning from scratch, growing efforts have been devoted to incorporating the benefits of pre-training. However, how to adaptively exploit the pre-trained knowledge for each incremental task while maintaining its generalizability remains an open question. In this work, we present an extensive analysis for continual learning on a pre-trained model (CLPM), and attribute the key challenge to a progressive overfitting problem. Observing that selectively reducing the learning rate can almost resolve this issue in the representation layer, we propose a simple but extremely effective approach named Slow Learner with Classifier Alignment (SLCA), which further improves the classification layer by modeling the class-wise distributions and aligning the classification layers in a post-hoc fashion. Across a variety of scenarios, our proposal provides substantial improvements for CLPM (e.g., up to 49.76%, 50.05%, 44.69% and 40.16% on Split CIFAR-100, Split ImageNet-R, Split CUB-200 and Split Cars-196, respectively), and thus outperforms state-of-the-art approaches by a large margin. Based on such a strong baseline, critical factors and promising directions are analyzed in-depth to facilitate subsequent research. Code has been made available at: https://github.com/GengDavid/SLCA.

By classifying infinite-width neural networks and identifying the *optimal* limit, Tensor Programs IV and V demonstrated a universal way, called muP, for *widthwise hyperparameter transfer*, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for *depthwise parametrizations* of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-muP that extends muP and show empirically it admits depthwise hyperparameter transfer. We identify *feature diversity* as a crucial factor in deep networks, and Depth-muP can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significantly better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl.

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Continual Pre-Training (CPT) has become a popular and effective method to apply strong foundation models to specific downstream tasks. In this work, we explore the learning dynamics throughout the CPT process for large language models. We specifically focus on how general and downstream domain performance evolves at each training step, with domain performance measured via validation losses. We have observed that the CPT loss curve fundamentally characterizes the transition from one curve to another hidden curve, and could be described by decoupling the effects of distribution shift and learning rate annealing. We derive a CPT scaling law that combines the two factors, enabling the prediction of loss at any (continual) training steps and across learning rate schedules (LRS) in CPT. Our formulation presents a comprehensive understanding of several critical factors in CPT, including loss potential, peak learning rate, training steps, replay ratio, etc. Moreover, our approach can be adapted to customize training hyper-parameters to different CPT goals such as balancing general and domain-specific performance. Extensive experiments demonstrate that our scaling law holds across various CPT datasets and training hyper-parameters.

Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we develop a novel theoretical framework for IL. Our main result is that IL closely approximates an optimization method known as implicit stochastic gradient descent (implicit SGD), which is distinct from the explicit SGD implemented by BP. Our results further show how the standard implementation of IL can be altered to better approximate implicit SGD. Our novel implementation considerably improves the stability of IL across learning rates, which is consistent with our theory, as a key property of implicit SGD is its stability. We provide extensive simulation results that further support our theoretical interpretations and also demonstrate IL achieves quicker convergence when trained with small mini-batches while matching the performance of BP for large mini-batches.

Normalization layers are a staple in state-of-the-art deep neural network architectures. They are widely believed to stabilize training, enable higher learning rate, accelerate convergence and improve generalization, though the reason for their effectiveness is still an active research topic. In this work, we challenge the commonly-held beliefs by showing that none of the perceived benefits is unique to normalization. Specifically, we propose fixed-update initialization (Fixup), an initialization motivated by solving the exploding and vanishing gradient problem at the beginning of training via properly rescaling a standard initialization. We find training residual networks with Fixup to be as stable as training with normalization -- even for networks with 10,000 layers. Furthermore, with proper regularization, Fixup enables residual networks without normalization to achieve state-of-the-art performance in image classification and machine translation.

While the phenomenon of grokking, i.e., delayed generalization, has been studied extensively, it remains an open problem whether there is a mathematical framework that characterizes what kind of features will emerge, how and in which conditions it happens, and is closely related to the gradient dynamics of the training, for complex structured inputs. We propose a novel framework, named Li_2, that captures three key stages for the grokking behavior of 2-layer nonlinear networks: (I) \textbf{L}azy learning, (II) \textbf{i}ndependent feature learning and (III) \textbf{i}nteractive feature learning. At the lazy learning stage, top layer overfits to random hidden representation and the model appears to memorize. Thanks to lazy learning and weight decay, the backpropagated gradient G_F from the top layer now carries information about the target label, with a specific structure that enables each hidden node to learn their representation independently. Interestingly, the independent dynamics follows exactly the gradient ascent of an energy function E, and its local maxima are precisely the emerging features. We study whether these local-optima induced features are generalizable, their representation power, and how they change on sample size, in group arithmetic tasks. When hidden nodes start to interact in the later stage of learning, we provably show how G_F changes to focus on missing features that need to be learned. Our study sheds lights on roles played by key hyperparameters such as weight decay, learning rate and sample sizes in grokking, leads to provable scaling laws of feature emergence, memorization and generalization, and reveals the underlying cause why recent optimizers such as Muon can be effective, from the first principles of gradient dynamics. Our analysis can be extended to multi-layer architectures.

This paper proposes a cold start recommendation model that integrates contrastive learning, aiming to solve the problem of performance degradation of recommendation systems in cold start scenarios due to the scarcity of user and item interaction data. The model dynamically adjusts the weights of key features through an adaptive feature selection module and effectively integrates user attributes, item meta-information, and contextual features by combining a multimodal feature fusion mechanism, thereby improving recommendation performance. In addition, the model introduces a contrastive learning mechanism to enhance the robustness and generalization ability of feature representation by constructing positive and negative sample pairs. Experiments are conducted on the MovieLens-1M dataset. The results show that the proposed model significantly outperforms mainstream recommendation methods such as Matrix Factorization, LightGBM, DeepFM, and AutoRec in terms of HR, NDCG, MRR, and Recall, especially in cold start scenarios. Ablation experiments further verify the key role of each module in improving model performance, and the learning rate sensitivity analysis shows that a moderate learning rate is crucial to the optimization effect of the model. This study not only provides a new solution to the cold start problem but also provides an important reference for the application of contrastive learning in recommendation systems. In the future, this model is expected to play a role in a wider range of scenarios, such as real-time recommendation and cross-domain recommendation.

Large language models encode impressively broad world knowledge in their parameters. However, the knowledge in static language models falls out of date, limiting the model's effective "shelf life." While online fine-tuning can reduce this degradation, we find that naively fine-tuning on a stream of documents leads to a low level of information uptake. We hypothesize that online fine-tuning does not sufficiently attend to important information. That is, the gradient signal from important tokens representing factual information is drowned out by the gradient from inherently noisy tokens, suggesting that a dynamic, context-aware learning rate may be beneficial. We therefore propose learning which tokens to upweight. We meta-train a small, autoregressive model to reweight the language modeling loss for each token during online fine-tuning, with the objective of maximizing the out-of-date base question-answering model's ability to answer questions about a document after a single weighted gradient step. We call this approach Context-aware Meta-learned Loss Scaling (CaMeLS). Across three different distributions of documents, our experiments find that CaMeLS provides substantially improved information uptake on streams of thousands of documents compared with standard fine-tuning and baseline heuristics for reweighting token losses.

Offline model-based optimization aims to maximize a black-box objective function with a static dataset of designs and their scores. In this paper, we focus on biological sequence design to maximize some sequence score. A recent approach employs bidirectional learning, combining a forward mapping for exploitation and a backward mapping for constraint, and it relies on the neural tangent kernel (NTK) of an infinitely wide network to build a proxy model. Though effective, the NTK cannot learn features because of its parametrization, and its use prevents the incorporation of powerful pre-trained Language Models (LMs) that can capture the rich biophysical information in millions of biological sequences. We adopt an alternative proxy model, adding a linear head to a pre-trained LM, and propose a linearization scheme. This yields a closed-form loss and also takes into account the biophysical information in the pre-trained LM. In addition, the forward mapping and the backward mapping play different roles and thus deserve different weights during sequence optimization. To achieve this, we train an auxiliary model and leverage its weak supervision signal via a bi-level optimization framework to effectively learn how to balance the two mappings. Further, by extending the framework, we develop the first learning rate adaptation module Adaptive-eta, which is compatible with all gradient-based algorithms for offline model-based optimization. Experimental results on DNA/protein sequence design tasks verify the effectiveness of our algorithm. Our code is available~https://anonymous.4open.science/r/BIB-ICLR2023-Submission/README.md{here.}

Federated learning (FL) is a trending training paradigm to utilize decentralized training data. FL allows clients to update model parameters locally for several epochs, then share them to a global model for aggregation. This training paradigm with multi-local step updating before aggregation exposes unique vulnerabilities to adversarial attacks. Adversarial training is a popular and effective method to improve the robustness of networks against adversaries. In this work, we formulate a general form of federated adversarial learning (FAL) that is adapted from adversarial learning in the centralized setting. On the client side of FL training, FAL has an inner loop to generate adversarial samples for adversarial training and an outer loop to update local model parameters. On the server side, FAL aggregates local model updates and broadcast the aggregated model. We design a global robust training loss and formulate FAL training as a min-max optimization problem. Unlike the convergence analysis in classical centralized training that relies on the gradient direction, it is significantly harder to analyze the convergence in FAL for three reasons: 1) the complexity of min-max optimization, 2) model not updating in the gradient direction due to the multi-local updates on the client-side before aggregation and 3) inter-client heterogeneity. We address these challenges by using appropriate gradient approximation and coupling techniques and present the convergence analysis in the over-parameterized regime. Our main result theoretically shows that the minimum loss under our algorithm can converge to epsilon small with chosen learning rate and communication rounds. It is noteworthy that our analysis is feasible for non-IID clients.

Amid mounting concern about the reliability and credibility of machine learning research, we present a principled framework for making robust and generalizable claims: the multiverse analysis. Our framework builds upon the multiverse analysis (Steegen et al., 2016) introduced in response to psychology's own reproducibility crisis. To efficiently explore high-dimensional and often continuous ML search spaces, we model the multiverse with a Gaussian Process surrogate and apply Bayesian experimental design. Our framework is designed to facilitate drawing robust scientific conclusions about model performance, and thus our approach focuses on exploration rather than conventional optimization. In the first of two case studies, we investigate disputed claims about the relative merit of adaptive optimizers. Second, we synthesize conflicting research on the effect of learning rate on the large batch training generalization gap. For the machine learning community, the multiverse analysis is a simple and effective technique for identifying robust claims, for increasing transparency, and a step toward improved reproducibility.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

Neural networks (NN) achieve remarkable results in various tasks, but lack key characteristics: interpretability, support for categorical features, and lightweight implementations suitable for edge devices. While ongoing efforts aim to address these challenges, Gradient Boosting Trees (GBT) inherently meet these requirements. As a result, GBTs have become the go-to method for supervised learning tasks in many real-world applications and competitions. However, their application in online learning scenarios, notably in reinforcement learning (RL), has been limited. In this work, we bridge this gap by introducing Gradient-Boosting RL (GBRL), a framework that extends the advantages of GBT to the RL domain. Using the GBRL framework, we implement various actor-critic algorithms and compare their performance with their NN counterparts. Inspired by shared backbones in NN we introduce a tree-sharing approach for policy and value functions with distinct learning rates, enhancing learning efficiency over millions of interactions. GBRL achieves competitive performance across a diverse array of tasks, excelling in domains with structured or categorical features. Additionally, we present a high-performance, GPU-accelerated implementation that integrates seamlessly with widely-used RL libraries (available at https://github.com/NVlabs/gbrl). GBRL expands the toolkit for RL practitioners, demonstrating the viability and promise of GBT within the RL paradigm, particularly in domains characterized by structured or categorical features.

Present-day federated learning (FL) systems deployed over edge networks consists of a large number of workers with high degrees of heterogeneity in data and/or computing capabilities, which call for flexible worker participation in terms of timing, effort, data heterogeneity, etc. To satisfy the need for flexible worker participation, we consider a new FL paradigm called "Anarchic Federated Learning" (AFL) in this paper. In stark contrast to conventional FL models, each worker in AFL has the freedom to choose i) when to participate in FL, and ii) the number of local steps to perform in each round based on its current situation (e.g., battery level, communication channels, privacy concerns). However, such chaotic worker behaviors in AFL impose many new open questions in algorithm design. In particular, it remains unclear whether one could develop convergent AFL training algorithms, and if yes, under what conditions and how fast the achievable convergence speed is. Toward this end, we propose two Anarchic Federated Averaging (AFA) algorithms with two-sided learning rates for both cross-device and cross-silo settings, which are named AFA-CD and AFA-CS, respectively. Somewhat surprisingly, we show that, under mild anarchic assumptions, both AFL algorithms achieve the best known convergence rate as the state-of-the-art algorithms for conventional FL. Moreover, they retain the highly desirable {\em linear speedup effect} with respect of both the number of workers and local steps in the new AFL paradigm. We validate the proposed algorithms with extensive experiments on real-world datasets.

Adversarial Training (AT) with Projected Gradient Descent (PGD) is an effective approach for improving the robustness of the deep neural networks. However, PGD AT has been shown to suffer from two main limitations: i) high computational cost, and ii) extreme overfitting during training that leads to reduction in model generalization. While the effect of factors such as model capacity and scale of training data on adversarial robustness have been extensively studied, little attention has been paid to the effect of a very important parameter in every network optimization on adversarial robustness: the learning rate. In particular, we hypothesize that effective learning rate scheduling during adversarial training can significantly reduce the overfitting issue, to a degree where one does not even need to adversarially train a model from scratch but can instead simply adversarially fine-tune a pre-trained model. Motivated by this hypothesis, we propose a simple yet very effective adversarial fine-tuning approach based on a slow start, fast decay learning rate scheduling strategy which not only significantly decreases computational cost required, but also greatly improves the accuracy and robustness of a deep neural network. Experimental results show that the proposed adversarial fine-tuning approach outperforms the state-of-the-art methods on CIFAR-10, CIFAR-100 and ImageNet datasets in both test accuracy and the robustness, while reducing the computational cost by 8-10times. Furthermore, a very important benefit of the proposed adversarial fine-tuning approach is that it enables the ability to improve the robustness of any pre-trained deep neural network without needing to train the model from scratch, which to the best of the authors' knowledge has not been previously demonstrated in research literature.

Transformers can capture long-range dependencies using self-attention, allowing tokens to attend to all others directly. However, stacking multiple attention layers leads to attention concentration. One natural way to address this issue is to use cross-layer attention, allowing information from earlier layers to be directly accessible to later layers. However, this approach is computationally expensive. To address this problem, we propose Transformer with residual value (ResFormer) which approximates cross-layer attention through adding a residual connection from the values of the the first layer to all subsequent layers. Based on this method, one variant is the Transformer with single layer value (SVFormer), where all layers share the same value embedding from first layer, reducing the KV cache by nearly 50%. Comprehensive empirical evidence demonstrates that ResFormer mitigates attention concentration problem in deeper layers and enhances representation across most layers, outperforming the vanilla Transformer, DenseFormer, and NeuTRENO in training error as well as downstream tasks. SVFormer trains significantly faster than the vanilla Transformer and performs better than other methods like GQA and CLA, with performance influenced by sequence length and cumulative learning rate.

The remarkable success of large language pretraining and the discovery of scaling laws signify a paradigm shift in machine learning. Notably, the primary objective has evolved from minimizing generalization error to reducing approximation error, and the most effective strategy has transitioned from regularization (in a broad sense) to scaling up models. This raises a critical question: Do the established principles that proved successful in the generalization-centric era remain valid in this new era of scaling? This paper examines several influential regularization-based principles that may no longer hold true in the scaling-centric, large language model (LLM) era. These principles include explicit L2 regularization and implicit regularization through small batch sizes and large learning rates. Additionally, we identify a new phenomenon termed ``scaling law crossover,'' where two scaling curves intersect at a certain scale, implying that methods effective at smaller scales may not generalize to larger ones. Together, these observations highlight two fundamental questions within this new paradigm: bullet Guiding Principles for Scaling: If regularization is no longer the primary guiding principle for model design, what new principles are emerging to guide scaling? bullet Model Comparison at Scale: How to reliably and effectively compare models at the scale where only a single experiment is feasible?

Federated learning is a powerful paradigm for large-scale machine learning, but it faces significant challenges due to unreliable network connections, slow communication, and substantial data heterogeneity across clients. FedAvg and SCAFFOLD are two prominent algorithms to address these challenges. In particular, FedAvg employs multiple local updates before communicating with a central server, while SCAFFOLD maintains a control variable on each client to compensate for ``client drift'' in its local updates. Various methods have been proposed to enhance the convergence of these two algorithms, but they either make impractical adjustments to the algorithmic structure or rely on the assumption of bounded data heterogeneity. This paper explores the utilization of momentum to enhance the performance of FedAvg and SCAFFOLD. When all clients participate in the training process, we demonstrate that incorporating momentum allows FedAvg to converge without relying on the assumption of bounded data heterogeneity even using a constant local learning rate. This is novel and fairly surprising as existing analyses for FedAvg require bounded data heterogeneity even with diminishing local learning rates. In partial client participation, we show that momentum enables SCAFFOLD to converge provably faster without imposing any additional assumptions. Furthermore, we use momentum to develop new variance-reduced extensions of FedAvg and SCAFFOLD, which exhibit state-of-the-art convergence rates. Our experimental results support all theoretical findings.

Different base language model families, such as Llama and Qwen, exhibit divergent behaviors during post-training with reinforcement learning (RL), especially on reasoning-intensive tasks. What makes a base language model suitable for reinforcement learning? Gaining deeper insight into this question is essential for developing RL-scalable foundation models of the next generation. In this work, we investigate how mid-training strategies shape RL dynamics, focusing on two representative model families: Qwen and Llama. Our study reveals that (1) high-quality mathematical corpora, such as MegaMath-Web-Pro, significantly improve both base model and RL performance, while existing alternatives (e.g., FineMath-4plus) fail to do so; (2) further adding QA-style data, particularly long chain-of-thought (CoT) reasoning examples, enhances RL outcomes, and instruction data further unlocks this effect; (3) while long-CoT improves reasoning depth, it can also induce verbosity of model responses and unstability of RL training, underscoring the importance of data formatting; (4) scaling mid-training consistently leads to stronger downstream RL performance. Building on these insights, we introduce a two-stage mid-training strategy, Stable-then-Decay, in which base models are first trained on 200B tokens with a constant learning rate, followed by 20B tokens across three CoT-focused branches with learning rate decay. This yields OctoThinker, a family of models demonstrating strong RL compatibility and closing the performance gap with more RL-friendly model families, i.e., Qwen. We hope our work will help shape pre-training strategies for foundation models in the RL era. To support further research, we release our open-source models along with a curated math reasoning-intensive corpus of over 70 billion tokens (i.e., MegaMath-Web-Pro-Max).

Recent advances in music foundation models have improved audio representation learning, yet their effectiveness across diverse musical traditions remains limited. We introduce CultureMERT-95M, a multi-culturally adapted foundation model developed to enhance cross-cultural music representation learning and understanding. To achieve this, we propose a two-stage continual pre-training strategy that integrates learning rate re-warming and re-decaying, enabling stable adaptation even with limited computational resources. Training on a 650-hour multi-cultural data mix, comprising Greek, Turkish, and Indian music traditions, results in an average improvement of 4.9% in ROC-AUC and AP across diverse non-Western music auto-tagging tasks, surpassing prior state-of-the-art, with minimal forgetting on Western-centric benchmarks. We further investigate task arithmetic, an alternative approach to multi-cultural adaptation that merges single-culture adapted models in the weight space. Task arithmetic performs on par with our multi-culturally trained model on non-Western auto-tagging tasks and shows no regression on Western datasets. Cross-cultural evaluation reveals that single-culture models transfer with varying effectiveness across musical traditions, whereas the multi-culturally adapted model achieves the best overall performance. To support research on world music representation learning, we publicly release CultureMERT-95M and CultureMERT-TA-95M, fostering the development of more culturally aware music foundation models.

In distributed deep learning with data parallelism, synchronizing gradients at each training step can cause a huge communication overhead, especially when many nodes work together to train large models. Local gradient methods, such as Local SGD, address this issue by allowing workers to compute locally for H steps without synchronizing with others, hence reducing communication frequency. While H has been viewed as a hyperparameter to trade optimization efficiency for communication cost, recent research indicates that setting a proper H value can lead to generalization improvement. Yet, selecting a proper H is elusive. This work proposes a theory-grounded method for determining H, named the Quadratic Synchronization Rule (QSR), which recommends dynamically setting H in proportion to 1{eta^2} as the learning rate eta decays over time. Extensive ImageNet experiments on ResNet and ViT show that local gradient methods with QSR consistently improve the test accuracy over other synchronization strategies. Compared with the standard data parallel training, QSR enables Local AdamW on ViT-B to cut the training time on 16 or 64 GPUs down from 26.7 to 20.2 hours or from 8.6 to 5.5 hours and, at the same time, achieves 1.16% or 0.84% higher top-1 validation accuracy.

As more non-AI experts use complex AI systems for daily tasks, there has been an increasing effort to develop methods that produce explanations of AI decision making that are understandable by non-AI experts. Towards this effort, leveraging higher-level concepts and producing concept-based explanations have become a popular method. Most concept-based explanations have been developed for classification techniques, and we posit that the few existing methods for sequential decision making are limited in scope. In this work, we first contribute a desiderata for defining concepts in sequential decision making settings. Additionally, inspired by the Protege Effect which states explaining knowledge often reinforces one's self-learning, we explore how concept-based explanations of an RL agent's decision making can in turn improve the agent's learning rate, as well as improve end-user understanding of the agent's decision making. To this end, we contribute a unified framework, State2Explanation (S2E), that involves learning a joint embedding model between state-action pairs and concept-based explanations, and leveraging such learned model to both (1) inform reward shaping during an agent's training, and (2) provide explanations to end-users at deployment for improved task performance. Our experimental validations, in Connect 4 and Lunar Lander, demonstrate the success of S2E in providing a dual-benefit, successfully informing reward shaping and improving agent learning rate, as well as significantly improving end user task performance at deployment time.

Layer normalization (LN) is a widely adopted deep learning technique especially in the era of foundation models. Recently, LN has been shown to be surprisingly effective in federated learning (FL) with non-i.i.d. data. However, exactly why and how it works remains mysterious. In this work, we reveal the profound connection between layer normalization and the label shift problem in federated learning. To understand layer normalization better in FL, we identify the key contributing mechanism of normalization methods in FL, called feature normalization (FN), which applies normalization to the latent feature representation before the classifier head. Although LN and FN do not improve expressive power, they control feature collapse and local overfitting to heavily skewed datasets, and thus accelerates global training. Empirically, we show that normalization leads to drastic improvements on standard benchmarks under extreme label shift. Moreover, we conduct extensive ablation studies to understand the critical factors of layer normalization in FL. Our results verify that FN is an essential ingredient inside LN to significantly improve the convergence of FL while remaining robust to learning rate choices, especially under extreme label shift where each client has access to few classes.

A modular design encourages neural models to disentangle and recombine different facets of knowledge to generalise more systematically to new tasks. In this work, we assume that each task is associated with a subset of latent discrete skills from a (potentially small) inventory. In turn, skills correspond to parameter-efficient (sparse / low-rank) model parameterisations. By jointly learning these and a task-skill allocation matrix, the network for each task is instantiated as the average of the parameters of active skills. To favour non-trivial soft partitions of skills across tasks, we experiment with a series of inductive biases, such as an Indian Buffet Process prior and a two-speed learning rate. We evaluate our latent-skill model on two main settings: 1) multitask reinforcement learning for grounded instruction following on 8 levels of the BabyAI platform; and 2) few-shot adaptation of pre-trained text-to-text generative models on CrossFit, a benchmark comprising 160 NLP tasks. We find that the modular design of a network significantly increases sample efficiency in reinforcement learning and few-shot generalisation in supervised learning, compared to baselines with fully shared, task-specific, or conditionally generated parameters where knowledge is entangled across tasks. In addition, we show how discrete skills help interpretability, as they yield an explicit hierarchy of tasks.

Training large deep neural networks on massive datasets is computationally very challenging. There has been recent surge in interest in using large batch stochastic optimization methods to tackle this issue. The most prominent algorithm in this line of research is LARS, which by employing layerwise adaptive learning rates trains ResNet on ImageNet in a few minutes. However, LARS performs poorly for attention models like BERT, indicating that its performance gains are not consistent across tasks. In this paper, we first study a principled layerwise adaptation strategy to accelerate training of deep neural networks using large mini-batches. Using this strategy, we develop a new layerwise adaptive large batch optimization technique called LAMB; we then provide convergence analysis of LAMB as well as LARS, showing convergence to a stationary point in general nonconvex settings. Our empirical results demonstrate the superior performance of LAMB across various tasks such as BERT and ResNet-50 training with very little hyperparameter tuning. In particular, for BERT training, our optimizer enables use of very large batch sizes of 32868 without any degradation of performance. By increasing the batch size to the memory limit of a TPUv3 Pod, BERT training time can be reduced from 3 days to just 76 minutes (Table 1). The LAMB implementation is available at https://github.com/tensorflow/addons/blob/master/tensorflow_addons/optimizers/lamb.py

This paper delves into the interplay between vision backbones and optimizers, unvealing an inter-dependent phenomenon termed \textbf{backbone-optimizer coupling bias} (BOCB). We observe that canonical CNNs, such as VGG and ResNet, exhibit a marked co-dependency with SGD families, while recent architectures like ViTs and ConvNeXt share a tight coupling with the adaptive learning rate ones. We further show that BOCB can be introduced by both optimizers and certain backbone designs and may significantly impact the pre-training and downstream fine-tuning of vision models. Through in-depth empirical analysis, we summarize takeaways on recommended optimizers and insights into robust vision backbone architectures. We hope this work can inspire the community to question long-held assumptions on backbones and optimizers, stimulate further explorations, and thereby contribute to more robust vision systems. The source code and models are publicly available at https://bocb-ai.github.io/.

This paper introduces Meta-PerSER, a novel meta-learning framework that personalizes Speech Emotion Recognition (SER) by adapting to each listener's unique way of interpreting emotion. Conventional SER systems rely on aggregated annotations, which often overlook individual subtleties and lead to inconsistent predictions. In contrast, Meta-PerSER leverages a Model-Agnostic Meta-Learning (MAML) approach enhanced with Combined-Set Meta-Training, Derivative Annealing, and per-layer per-step learning rates, enabling rapid adaptation with only a few labeled examples. By integrating robust representations from pre-trained self-supervised models, our framework first captures general emotional cues and then fine-tunes itself to personal annotation styles. Experiments on the IEMOCAP corpus demonstrate that Meta-PerSER significantly outperforms baseline methods in both seen and unseen data scenarios, highlighting its promise for personalized emotion recognition.

Existing learning rate schedules that do not require specification of the optimization stopping step T are greatly out-performed by learning rate schedules that depend on T. We propose an approach that avoids the need for this stopping time by eschewing the use of schedules entirely, while exhibiting state-of-the-art performance compared to schedules across a wide family of problems ranging from convex problems to large-scale deep learning problems. Our Schedule-Free approach introduces no additional hyper-parameters over standard optimizers with momentum. Our method is a direct consequence of a new theory we develop that unifies scheduling and iterate averaging. An open source implementation of our method is available (https://github.com/facebookresearch/schedule_free).

Transformers consist of diverse building blocks, such as embedding layers, normalization layers, self-attention mechanisms, and point-wise feedforward networks. Thus, understanding the differences and interactions among these blocks is important. In this paper, we uncover a clear Sharpness Disparity across these blocks, which emerges early in training and intriguingly persists throughout the training process. Motivated by this finding, we propose Blockwise Learning Rate (LR), a strategy that tailors the LR to each block's sharpness, accelerating large language model (LLM) pre-training. By integrating Blockwise LR into AdamW, we consistently achieve lower terminal loss and nearly 2times speedup compared to vanilla AdamW. We demonstrate this acceleration across GPT-2 and LLaMA, with model sizes ranging from 0.12B to 1.1B and datasets of OpenWebText and MiniPile. Finally, we incorporate Blockwise LR into Adam-mini (Zhang et al., 2024), a recently proposed memory-efficient variant of Adam, achieving a combined 2times speedup and 2times memory saving. These results underscore the potential of exploiting the sharpness disparity to improve LLM training.

Large language models (LLMs) are routinely pre-trained on billions of tokens, only to restart the process over again once new data becomes available. A much cheaper and more efficient solution would be to enable the continual pre-training of these models, i.e. updating pre-trained models with new data instead of re-training them from scratch. However, the distribution shift induced by novel data typically results in degraded performance on past data. Taking a step towards efficient continual pre-training, in this work, we examine the effect of different warm-up strategies. Our hypothesis is that the learning rate must be re-increased to improve compute efficiency when training on a new dataset. We study the warmup phase of models pre-trained on the Pile (upstream data, 300B tokens) as we continue to pre-train on SlimPajama (downstream data, 297B tokens), following a linear warmup and cosine decay schedule. We conduct all experiments on the Pythia 410M language model architecture and evaluate performance through validation perplexity. We experiment with different pre-training checkpoints, various maximum learning rates, and various warmup lengths. Our results show that while rewarming models first increases the loss on upstream and downstream data, in the longer run it improves the downstream performance, outperforming models trained from scratchx2013even for a large downstream dataset.

Large language models (LLMs) are notoriously memory-intensive during training, particularly with the popular AdamW optimizer. This memory burden necessitates using more or higher-end GPUs or reducing batch sizes, limiting training scalability and throughput. To address this, various memory-efficient optimizers have been proposed to reduce optimizer memory usage. However, they face critical challenges: (i) reliance on costly SVD operations; (ii) significant performance trade-offs compared to AdamW; and (iii) still substantial optimizer memory overhead to maintain competitive performance. In this work, we identify that AdamW's learning rate adaptation rule can be effectively coarsened as a structured learning rate update. Based on this insight, we propose Approximated Gradient Scaling for Memory-Efficient LLM Optimization (APOLLO), which approximates learning rate scaling using an auxiliary low-rank optimizer state based on pure random projection. This structured learning rate update rule makes APOLLO highly tolerant to further memory reductions while delivering comparable pre-training performance. Even its rank-1 variant, APOLLO-Mini, achieves superior pre-training performance compared to AdamW with SGD-level memory costs. Extensive experiments demonstrate that the APOLLO series performs on-par with or better than AdamW, while achieving greater memory savings by nearly eliminating the optimization states of AdamW. These savings provide significant system-level benefits: (1) Enhanced Throughput: 3x throughput on an 8xA100-80GB setup compared to AdamW by supporting 4x larger batch sizes. (2) Improved Model Scalability: Pre-training LLaMA-13B with naive DDP on A100-80GB GPUs without system-level optimizations. (3) Low-End GPU Friendly Pre-training: Pre-training LLaMA-7B on a single GPU using less than 12 GB of memory with weight quantization.

Scale has become a main ingredient in obtaining strong machine learning models. As a result, understanding a model's scaling properties is key to effectively designing both the right training setup as well as future generations of architectures. In this work, we argue that scale and training research has been needlessly complex due to reliance on the cosine schedule, which prevents training across different lengths for the same model size. We investigate the training behavior of a direct alternative - constant learning rate and cooldowns - and find that it scales predictably and reliably similar to cosine. Additionally, we show that stochastic weight averaging yields improved performance along the training trajectory, without additional training costs, across different scales. Importantly, with these findings we demonstrate that scaling experiments can be performed with significantly reduced compute and GPU hours by utilizing fewer but reusable training runs.

Adaptive gradient methods, e.g. Adam, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to suffer from compromised generalization ability compared with stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second raw moment estimate term with its momentumized version in Adam can resolve the issue. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a more favorable option for learning rate scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing much better. We further develop a theory to back up the improvement in generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance and superior training stability consistently.

Hyperspectral image classification remains a challenging task due to the high dimensionality of spectral data, significant inter-band redundancy, and the limited availability of annotated samples. While recent transformer-based models have improved the global modeling of spectral-spatial dependencies, their scalability and adaptability under label-scarce conditions remain limited. In this work, we propose LoLA-SpecViT(Low-rank adaptation Local Attention Spectral Vision Transformer), a lightweight spectral vision transformer that addresses these limitations through a parameter-efficient architecture tailored to the unique characteristics of hyperspectral imagery. Our model combines a 3D convolutional spectral front-end with local window-based self-attention, enhancing both spectral feature extraction and spatial consistency while reducing computational complexity. To further improve adaptability, we integrate low-rank adaptation (LoRA) into attention and projection layers, enabling fine-tuning with over 80\% fewer trainable parameters. A novel cyclical learning rate scheduler modulates LoRA adaptation strength during training, improving convergence and generalisation. Extensive experiments on three benchmark datasets WHU-Hi LongKou, WHU-Hi HongHu, and Salinas demonstrate that LoLA-SpecViT consistently outperforms state-of-the-art baselines, achieving up to 99.91\% accuracy with substantially fewer parameters and enhanced robustness under low-label regimes. The proposed framework provides a scalable and generalizable solution for real-world HSI applications in agriculture, environmental monitoring, and remote sensing analytics. Our code is available in the following https://github.com/FadiZidiDz/LoLA-SpecViT{GitHub Repository}.

Recent studies suggest that with sufficiently wide models, most SGD solutions can, up to permutation, converge into the same basin. This phenomenon, known as the model re-basin regime, has significant implications for model averaging by ensuring the linear mode connectivity. However, current re-basin strategies are ineffective in many scenarios due to a lack of comprehensive understanding of underlying mechanisms. Addressing this gap, this paper provides novel insights into understanding and improving the standard practice. Firstly, we decompose re-normalization into rescaling and reshift, uncovering that rescaling plays a crucial role in re-normalization while re-basin performance is sensitive to shifts in model activation. The finding calls for a more nuanced handling of the activation shift. Secondly, we identify that the merged model suffers from the issue of activation collapse and magnitude collapse. Varying the learning rate, weight decay, and initialization method can mitigate the issues and improve model performance. Lastly, we propose a new perspective to unify the re-basin and pruning, under which a lightweight yet effective post-pruning technique is derived, which can significantly improve the model performance after pruning. Our implementation is available at https://github.com/XingyuQu/rethink-re-basin.

In this paper, we study the implicit regularization of stochastic gradient descent (SGD) through the lens of {\em dynamical stability} (Wu et al., 2018). We start by revising existing stability analyses of SGD, showing how the Frobenius norm and trace of Hessian relate to different notions of stability. Notably, if a global minimum is linearly stable for SGD, then the trace of Hessian must be less than or equal to 2/eta, where eta denotes the learning rate. By contrast, for gradient descent (GD), the stability imposes a similar constraint but only on the largest eigenvalue of Hessian. We then turn to analyze the generalization properties of these stable minima, focusing specifically on two-layer ReLU networks and diagonal linear networks. Notably, we establish the {\em equivalence} between these metrics of sharpness and certain parameter norms for the two models, which allows us to show that the stable minima of SGD provably generalize well. By contrast, the stability-induced regularization of GD is provably too weak to ensure satisfactory generalization. This discrepancy provides an explanation of why SGD often generalizes better than GD. Note that the learning rate (LR) plays a pivotal role in the strength of stability-induced regularization. As the LR increases, the regularization effect becomes more pronounced, elucidating why SGD with a larger LR consistently demonstrates superior generalization capabilities. Additionally, numerical experiments are provided to support our theoretical findings.

Deep learning practitioners often operate on a computational and monetary budget. Thus, it is critical to design optimization algorithms that perform well under any budget. The linear learning rate schedule is considered the best budget-aware schedule, as it outperforms most other schedules in the low budget regime. On the other hand, learning rate schedules -- such as the 30-60-90 step schedule -- are known to achieve high performance when the model can be trained for many epochs. Yet, it is often not known a priori whether one's budget will be large or small; thus, the optimal choice of learning rate schedule is made on a case-by-case basis. In this paper, we frame the learning rate schedule selection problem as a combination of i) selecting a profile (i.e., the continuous function that models the learning rate schedule), and ii) choosing a sampling rate (i.e., how frequently the learning rate is updated/sampled from this profile). We propose a novel profile and sampling rate combination called the Reflected Exponential (REX) schedule, which we evaluate across seven different experimental settings with both SGD and Adam optimizers. REX outperforms the linear schedule in the low budget regime, while matching or exceeding the performance of several state-of-the-art learning rate schedules (linear, step, exponential, cosine, step decay on plateau, and OneCycle) in both high and low budget regimes. Furthermore, REX requires no added computation, storage, or hyperparameters.

Generative Adversarial Networks (GANs) excel at creating realistic images with complex models for which maximum likelihood is infeasible. However, the convergence of GAN training has still not been proved. We propose a two time-scale update rule (TTUR) for training GANs with stochastic gradient descent on arbitrary GAN loss functions. TTUR has an individual learning rate for both the discriminator and the generator. Using the theory of stochastic approximation, we prove that the TTUR converges under mild assumptions to a stationary local Nash equilibrium. The convergence carries over to the popular Adam optimization, for which we prove that it follows the dynamics of a heavy ball with friction and thus prefers flat minima in the objective landscape. For the evaluation of the performance of GANs at image generation, we introduce the "Fr\'echet Inception Distance" (FID) which captures the similarity of generated images to real ones better than the Inception Score. In experiments, TTUR improves learning for DCGANs and Improved Wasserstein GANs (WGAN-GP) outperforming conventional GAN training on CelebA, CIFAR-10, SVHN, LSUN Bedrooms, and the One Billion Word Benchmark.

Despite recent progress in optimal hyperparameter transfer under model and dataset scaling, no unifying explanatory principle has been established. Using the Scion optimizer, we discover that joint optimal scaling across model and dataset sizes is governed by a single invariant: the operator norm of the output layer. Across models with up to 1.3B parameters trained on up to 138B tokens, the optimal learning rate/batch size pair (eta^{ast}, B^{ast}) consistently has the same operator norm value - a phenomenon we term norm transfer. This constant norm condition is necessary but not sufficient: while for each dataset size, multiple (eta, B) reach the optimal norm, only a unique (eta^{ast}, B^{ast}) achieves the best loss. As a sufficient condition, we provide the first measurement of (eta^{ast}, B^{ast}) scaling with dataset size for Scion, and find that the scaling rules are consistent with those of the Adam optimizer. Tuning per-layer-group learning rates also improves model performance, with the output layer being the most sensitive and hidden layers benefiting from lower learning rates. We provide practical insights on norm-guided optimal scaling and release our Distributed Scion (Disco) implementation with logs from over two thousand runs to support research on LLM training dynamics at scale.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the muParam (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

Empirical scaling laws prescribe how to allocate parameters, data, and compute, while maximal-update parameterization (muP) enables learning-rate transfer across widths by equalizing early-time update magnitudes. However, in modern scale-invariant architectures, training quickly enters an optimizer-governed steady state where normalization layers create backward scale sensitivity and the effective learning rate becomes width dependent, degrading muP transfer. We address this by introducing a weight-decay scaling rule for AdamW that preserves sublayer gain across widths. Empirically, the singular-value spectrum of each matrix parameter scales in norm as eta/lambda with an approximately invariant shape; under width scaling d, we observe that the top singular value scales approximately as eta/lambdacdot d^{0.75}. Combining this observation with the muP learning-rate rule eta_2propto d^{-1} for matrix-like parameters implies an empirical weight-decay scaling rule lambda_2propto d that approximately keeps sublayer gains width invariant. Together with vector-like parameters trained at eta_1=Theta_d(1) and lambda_1=0, this yields zero-shot transfer of both learning rate and weight decay from proxy to target widths, removing per-width sweeps. We validate the rule on LLaMA-style Transformers and in a minimal synthetic setting, and we provide a simple diagnostic, matching top singular values, to check sublayer-gain invariance. Our results extend muP beyond the near-init regime by explicitly controlling steady-state scales set by the optimizer, offering a practical recipe for width-robust hyperparameter transfer under AdamW.

Conventional wisdom dictates that small batch sizes make language model pretraining and fine-tuning unstable, motivating gradient accumulation, which trades off the number of optimizer steps for a proportional increase in batch size. While it is common to decrease the learning rate for smaller batch sizes, other hyperparameters are often held fixed. In this work, we revisit small batch sizes all the way down to batch size one, and we propose a rule for scaling Adam hyperparameters to small batch sizes. We find that small batch sizes (1) train stably, (2) are consistently more robust to hyperparameter choices, (3) achieve equal or better per-FLOP performance than larger batch sizes, and (4) notably enable stable language model training with vanilla SGD, even without momentum, despite storing no optimizer state. Building on these results, we provide practical recommendations for selecting a batch size and setting optimizer hyperparameters. We further recommend against gradient accumulation unless training on multiple devices with multiple model replicas, bottlenecked by inter-device bandwidth.

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, Lion (Evo\textbf{Lved Sign Momentum}). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3% zero-shot and 91.1% fine-tuning accuracy on ImageNet, surpassing the previous best results by 2% and 0.1%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant. The implementation of Lion is publicly available.

We present the Large Language Model from Power Law Decoder Representations (PLDR-LLM), a language model that leverages non-linear and linear transformations through Power Law Graph Attention mechanism to generate well-defined deductive and inductive outputs. We pretrain the PLDR-LLMs of varying layer sizes with a small batch size of 32 and sim8B tokens from the RefinedWeb dataset, and show that they achieve competitive performance in zero-shot and few-shot settings compared to scaled dot-product LLMs of similar model size reported in the literature. We show that deductive outputs of PLDR-LLMs can be used to compare model characteristics or improve the performance by introducing the Directed Acyclic Graph (DAG) loss as a metric and regularizer. Our results indicate that the initial maximum learning rate and warm-up steps have a lasting impact on deductive outputs throughout the pretraining. We provide a detailed description of PLDR-LLM architecture, its implementation and the pretraining procedure.

Robust and effective scaling of models from small to large width typically requires the precise adjustment of many algorithmic and architectural details, such as parameterization and optimizer choices. In this work, we propose a new perspective on parameterization by investigating a key assumption in prior work about the alignment between parameters and data and derive new theoretical results under weaker assumptions and a broader set of optimizers. Our extensive empirical investigation includes tens of thousands of models trained with all combinations of three optimizers, four parameterizations, several alignment assumptions, more than a dozen learning rates, and fourteen model sizes up to 26.8B parameters. We find that the best learning rate scaling prescription would often have been excluded by the assumptions in prior work. Our results show that all parameterizations, not just maximal update parameterization (muP), can achieve hyperparameter transfer; moreover, our novel per-layer learning rate prescription for standard parameterization outperforms muP. Finally, we demonstrate that an overlooked aspect of parameterization, the epsilon parameter in Adam, must be scaled correctly to avoid gradient underflow and propose Adam-atan2, a new numerically stable, scale-invariant version of Adam that eliminates the epsilon hyperparameter entirely.

The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.

We participated in the SynthRAD2025 challenge (Tasks 1 and 2) with a unified pipeline for synthetic CT (sCT) generation from MRI and CBCT, implemented using the KonfAI framework. Our model is a 2.5D U-Net++ with a ResNet-34 encoder, trained jointly across anatomical regions and fine-tuned per region. The loss function combined pixel-wise L1 loss with IMPACT-Synth, a perceptual loss derived from SAM and TotalSegmentator to enhance structural fidelity. Training was performed using AdamW (initial learning rate = 0.001, halved every 25k steps) on patch-based, normalized, body-masked inputs (320x320 for MRI, 256x256 for CBCT), with random flipping as the only augmentation. No post-processing was applied. Final predictions leveraged test-time augmentation and five-fold ensembling. The best model was selected based on validation MAE. Two registration strategies were evaluated: (i) Elastix with mutual information, consistent with the challenge pipeline, and (ii) IMPACT, a feature-based similarity metric leveraging pretrained segmentation networks. On the local test sets, IMPACT-based registration achieved more accurate and anatomically consistent alignments than mutual-information-based registration, resulting in improved sCT synthesis with lower MAE and more realistic anatomical structures. On the public validation set, however, models trained with Elastix-aligned data achieved higher scores, reflecting a registration bias favoring alignment strategies consistent with the evaluation pipeline. This highlights how registration errors can propagate into supervised learning, influencing both training and evaluation, and potentially inflating performance metrics at the expense of anatomical fidelity. By promoting anatomically consistent alignment, IMPACT helps mitigate this bias and supports the development of more robust and generalizable sCT synthesis models.

Model quantization is widely applied for compressing and accelerating deep neural networks (DNNs). However, conventional Quantization-Aware Training (QAT) focuses on training DNNs with uniform bit-width. The bit-width settings vary across different hardware and transmission demands, which induces considerable training and storage costs. Hence, the scheme of one-shot joint training multiple precisions is proposed to address this issue. Previous works either store a larger FP32 model to switch between different precision models for higher accuracy or store a smaller INT8 model but compromise accuracy due to using shared quantization parameters. In this paper, we introduce the Double Rounding quantization method, which fully utilizes the quantized representation range to accomplish nearly lossless bit-switching while reducing storage by using the highest integer precision instead of full precision. Furthermore, we observe a competitive interference among different precisions during one-shot joint training, primarily due to inconsistent gradients of quantization scales during backward propagation. To tackle this problem, we propose an Adaptive Learning Rate Scaling (ALRS) technique that dynamically adapts learning rates for various precisions to optimize the training process. Additionally, we extend our Double Rounding to one-shot mixed precision training and develop a Hessian-Aware Stochastic Bit-switching (HASB) strategy. Experimental results on the ImageNet-1K classification demonstrate that our methods have enough advantages to state-of-the-art one-shot joint QAT in both multi-precision and mixed-precision. We also validate the feasibility of our method on detection and segmentation tasks, as well as on LLMs task. Our codes are available at https://github.com/haiduo/Double-Rounding.

As language models have scaled both their number of parameters and pretraining dataset sizes, the computational cost for pretraining has become intractable except for the most well-resourced teams. This increasing cost makes it ever more important to be able to reuse a model after it has completed pretraining; allowing for a model's abilities to further improve without needing to train from scratch. In this work, we detail a set of guidelines that cover how to design efficacious data distributions and learning rate schedules for continued pretraining of language models. When applying these findings within a continued pretraining run on top of a well-trained 15B parameter model, we show an improvement of 9\% in average model accuracy compared to the baseline of continued training on the pretraining set. The resulting recipe provides a practical starting point with which to begin developing language models through reuse rather than retraining.

We present VoloGAN, an adversarial domain adaptation network that translates synthetic RGB-D images of a high-quality 3D model of a person, into RGB-D images that could be generated with a consumer depth sensor. This system is especially useful to generate high amount training data for single-view 3D reconstruction algorithms replicating the real-world capture conditions, being able to imitate the style of different sensor types, for the same high-end 3D model database. The network uses a CycleGAN framework with a U-Net architecture for the generator and a discriminator inspired by SIV-GAN. We use different optimizers and learning rate schedules to train the generator and the discriminator. We further construct a loss function that considers image channels individually and, among other metrics, evaluates the structural similarity. We demonstrate that CycleGANs can be used to apply adversarial domain adaptation of synthetic 3D data to train a volumetric video generator model having only few training samples.

We analyze the convergence of (stochastic) gradient descent algorithm for learning a convolutional filter with Rectified Linear Unit (ReLU) activation function. Our analysis does not rely on any specific form of the input distribution and our proofs only use the definition of ReLU, in contrast with previous works that are restricted to standard Gaussian input. We show that (stochastic) gradient descent with random initialization can learn the convolutional filter in polynomial time and the convergence rate depends on the smoothness of the input distribution and the closeness of patches. To the best of our knowledge, this is the first recovery guarantee of gradient-based algorithms for convolutional filter on non-Gaussian input distributions. Our theory also justifies the two-stage learning rate strategy in deep neural networks. While our focus is theoretical, we also present experiments that illustrate our theoretical findings.

Background: Deep learning models are typically trained using stochastic gradient descent or one of its variants. These methods update the weights using their gradient, estimated from a small fraction of the training data. It has been observed that when using large batch sizes there is a persistent degradation in generalization performance - known as the "generalization gap" phenomena. Identifying the origin of this gap and closing it had remained an open problem. Contributions: We examine the initial high learning rate training phase. We find that the weight distance from its initialization grows logarithmically with the number of weight updates. We therefore propose a "random walk on random landscape" statistical model which is known to exhibit similar "ultra-slow" diffusion behavior. Following this hypothesis we conducted experiments to show empirically that the "generalization gap" stems from the relatively small number of updates rather than the batch size, and can be completely eliminated by adapting the training regime used. We further investigate different techniques to train models in the large-batch regime and present a novel algorithm named "Ghost Batch Normalization" which enables significant decrease in the generalization gap without increasing the number of updates. To validate our findings we conduct several additional experiments on MNIST, CIFAR-10, CIFAR-100 and ImageNet. Finally, we reassess common practices and beliefs concerning training of deep models and suggest they may not be optimal to achieve good generalization.

The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon's emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-order momentum statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon's conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that strategically distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both Adam and Muon, achieving 21.74% better training efficiency than Adam and 11.31% improvement over Muon on 1.1 B pretraining setting, while maintaining a comparable memory footprint to Muon. Our findings suggest that orthogonalization and adaptive learning rates are complementary rather than competing approaches, opening new avenues for optimizer design in large-scale deep learning.

In this paper, we show that Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021) leads to suboptimal finetuning of models with large width (embedding dimension). This is due to the fact that adapter matrices A and B in LoRA are updated with the same learning rate. Using scaling arguments for large width networks, we demonstrate that using the same learning rate for A and B does not allow efficient feature learning. We then show that this suboptimality of LoRA can be corrected simply by setting different learning rates for the LoRA adapter matrices A and B with a well-chosen ratio. We call this proposed algorithm LoRA+. In our extensive experiments, LoRA+ improves performance (1-2 % improvements) and finetuning speed (up to sim 2X SpeedUp), at the same computational cost as LoRA.

State-of-the-art LLMs are powered by scaling -- scaling model size, dataset size and cluster size. It is economically infeasible to extensively tune hyperparameter for the largest runs. Instead, approximately optimal hyperparameters must be inferred or transferred from smaller experiments. Hyperparameter transfer across model sizes has been studied in Yang et al. However, hyperparameter transfer across dataset size -- or token horizon -- has not been studied yet. To remedy this we conduct a large scale empirical study on how optimal learning rate (LR) depends on token horizon in LLM training. We first demonstrate that the optimal LR changes significantly with token horizon -- longer training necessitates smaller LR. Secondly we demonstrate the the optimal LR follows a scaling law, and that the optimal LR for longer horizons can be accurately estimated from shorter horizons via such scaling laws. We also provide a rule-of-thumb for transferring LR across token horizons with zero overhead over current practices. Lastly we provide evidence that LLama-1 used too high LR, and estimate the performance hit from this. We thus argue that hyperparameter transfer across data size is an important and overlooked component of LLM training.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

The choice of batch sizes in stochastic gradient optimizers is critical for model training. However, the practice of varying batch sizes throughout the training process is less explored compared to other hyperparameters. We investigate adaptive batch size strategies derived from adaptive sampling methods, traditionally applied only in stochastic gradient descent. Given the significant interplay between learning rates and batch sizes, and considering the prevalence of adaptive gradient methods in deep learning, we emphasize the need for adaptive batch size strategies in these contexts. We introduce AdAdaGrad and its scalar variant AdAdaGradNorm, which incrementally increase batch sizes during training, while model updates are performed using AdaGrad and AdaGradNorm. We prove that AdaGradNorm converges with high probability at a rate of O(1/K) for finding a first-order stationary point of smooth nonconvex functions within K iterations. AdaGrad also demonstrates similar convergence properties when integrated with a novel coordinate-wise variant of our adaptive batch size strategies. Our theoretical claims are supported by numerical experiments on various image classification tasks, highlighting the enhanced adaptability of progressive batching protocols in deep learning and the potential of such adaptive batch size strategies with adaptive gradient optimizers in large-scale model training.

Deep learning thrives with large neural networks and large datasets. However, larger networks and larger datasets result in longer training times that impede research and development progress. Distributed synchronous SGD offers a potential solution to this problem by dividing SGD minibatches over a pool of parallel workers. Yet to make this scheme efficient, the per-worker workload must be large, which implies nontrivial growth in the SGD minibatch size. In this paper, we empirically show that on the ImageNet dataset large minibatches cause optimization difficulties, but when these are addressed the trained networks exhibit good generalization. Specifically, we show no loss of accuracy when training with large minibatch sizes up to 8192 images. To achieve this result, we adopt a hyper-parameter-free linear scaling rule for adjusting learning rates as a function of minibatch size and develop a new warmup scheme that overcomes optimization challenges early in training. With these simple techniques, our Caffe2-based system trains ResNet-50 with a minibatch size of 8192 on 256 GPUs in one hour, while matching small minibatch accuracy. Using commodity hardware, our implementation achieves ~90% scaling efficiency when moving from 8 to 256 GPUs. Our findings enable training visual recognition models on internet-scale data with high efficiency.

Large language models (LLMs) are routinely pre-trained on billions of tokens, only to start the process over again once new data becomes available. A much more efficient solution is to continually pre-train these models, saving significant compute compared to re-training. However, the distribution shift induced by new data typically results in degraded performance on previous data or poor adaptation to the new data. In this work, we show that a simple and scalable combination of learning rate (LR) re-warming, LR re-decaying, and replay of previous data is sufficient to match the performance of fully re-training from scratch on all available data, as measured by final loss and language model (LM) evaluation benchmarks. Specifically, we show this for a weak but realistic distribution shift between two commonly used LLM pre-training datasets (EnglishrightarrowEnglish) and a stronger distribution shift (EnglishrightarrowGerman) at the 405M parameter model scale with large dataset sizes (hundreds of billions of tokens). Selecting the weak but realistic shift for larger-scale experiments, we also find that our continual learning strategies match the re-training baseline for a 10B parameter LLM. Our results demonstrate that LLMs can be successfully updated via simple and scalable continual learning strategies, matching the re-training baseline using only a fraction of the compute. Finally, inspired by previous work, we propose alternatives to the cosine learning rate schedule that help circumvent forgetting induced by LR re-warming and that are not bound to a fixed token budget.

Xmodel-2 is a 1.2-billion-parameter large language model designed specifically for reasoning tasks. Its architecture enables different model scales to share a unified set of hyperparameters, allowing for extensive experimentation on smaller models and seamless transfer of optimal configurations to larger models. To maximize training efficiency and stability, Xmodel-2 employs the WSD learning rate scheduler from MiniCPM. Pretrained on 1.5 trillion tokens from diverse sources, Xmodel-2 achieves state-of-the-art performance in complex reasoning and agent-based tasks, while maintaining low training costs. These results highlight the potential of efficient model design and training strategies in advancing reasoning capabilities. Model checkpoints and code are publicly available on GitHub at https://github.com/XiaoduoAILab/Xmodel-2

While post-training quantization is widely adopted for efficient deployment of large language models, the mechanisms underlying quantization robustness remain unclear. We conduct a comprehensive analysis of quantization degradation across open-source language model training trajectories up to 32B parameters and 15T training tokens to accurately assess the relationship between training dynamics and quantization performance. Our key finding is that quantization errors in large-scale training runs are driven by a complex interplay between learning rate and other training hyperparameters. Specifically, once learning rates decay, validation loss and quantization error diverge, largely independent of training data scale. To investigate interventions on the training dynamics and identify specific configurations that can modulate quantization robustness favorably, we train our own models in controlled experiments up to 100B tokens. Our results challenge the assumption that increasing dataset scale inherently compromises quantization effectiveness, demonstrating instead that strategic training hyperparameter interventions can improve quantization quality at scale.

Generative data augmentation, which scales datasets by obtaining fake labeled examples from a trained conditional generative model, boosts classification performance in various learning tasks including (semi-)supervised learning, few-shot learning, and adversarially robust learning. However, little work has theoretically investigated the effect of generative data augmentation. To fill this gap, we establish a general stability bound in this not independently and identically distributed (non-i.i.d.) setting, where the learned distribution is dependent on the original train set and generally not the same as the true distribution. Our theoretical result includes the divergence between the learned distribution and the true distribution. It shows that generative data augmentation can enjoy a faster learning rate when the order of divergence term is o(maxleft( log(m)beta_m, 1 / m)right), where m is the train set size and beta_m is the corresponding stability constant. We further specify the learning setup to the Gaussian mixture model and generative adversarial nets. We prove that in both cases, though generative data augmentation does not enjoy a faster learning rate, it can improve the learning guarantees at a constant level when the train set is small, which is significant when the awful overfitting occurs. Simulation results on the Gaussian mixture model and empirical results on generative adversarial nets support our theoretical conclusions. Our code is available at https://github.com/ML-GSAI/Understanding-GDA.

Training large neural networks is known to be time-consuming, with the learning duration taking days or even weeks. To address this problem, large-batch optimization was introduced. This approach demonstrated that scaling mini-batch sizes with appropriate learning rate adjustments can speed up the training process by orders of magnitude. While long training time was not typically a major issue for model-free deep offline RL algorithms, recently introduced Q-ensemble methods achieving state-of-the-art performance made this issue more relevant, notably extending the training duration. In this work, we demonstrate how this class of methods can benefit from large-batch optimization, which is commonly overlooked by the deep offline RL community. We show that scaling the mini-batch size and naively adjusting the learning rate allows for (1) a reduced size of the Q-ensemble, (2) stronger penalization of out-of-distribution actions, and (3) improved convergence time, effectively shortening training duration by 3-4x times on average.

Large-scale transformer models have become the de-facto architectures for various machine learning applications, e.g., CV and NLP. However, those large models also introduce prohibitive training costs. To mitigate this issue, we propose a novel random and layerwise token dropping method (random-LTD), which skips the computation of a subset of the input tokens at all middle layers. Particularly, random-LTD achieves considerable speedups and comparable accuracy as the standard training baseline. Compared to other token dropping methods, random-LTD does not require (1) any importance score-based metrics, (2) any special token treatment (e.g., [CLS]), and (3) many layers in full sequence length training except the first and the last layers. Besides, a new LayerToken learning rate schedule is proposed for pretraining problems that resolve the heavy tuning requirement for our proposed training mechanism. Finally, we demonstrate that random-LTD can be applied to broader applications, including GPT and BERT pretraining as well as ViT and GPT finetuning tasks. Our results show that random-LTD can save about 33.3% theoretical compute cost and 25.6% wall-clock training time while achieving similar zero-shot evaluations on GPT-31.3B as compared to baseline.

Efficient LLM pre-training requires well-tuned hyperparameters (HPs), including learning rate {\eta} and weight decay {\lambda}. We study scaling laws for HPs: formulas for how to scale HPs as we scale model size N, dataset size D, and batch size B. Recent work suggests the AdamW timescale, B/({\eta}{\lambda}D), should remain constant across training settings, and we verify the implication that optimal {\lambda} scales linearly with B, for a fixed N,D. However, as N,D scale, we show the optimal timescale obeys a precise power law in the tokens-per-parameter ratio, D/N. This law thus provides a method to accurately predict {\lambda}opt in advance of large-scale training. We also study scaling laws for optimal batch size Bopt (the B enabling lowest loss at a given N,D) and critical batch size Bcrit (the B beyond which further data parallelism becomes ineffective). In contrast with prior work, we find both Bopt and Bcrit scale as power laws in D, independent of model size, N. Finally, we analyze how these findings inform the real-world selection of Pareto-optimal N and D under dual training time and compute objectives.

We study compute efficiency of LLM training when using different parameterizations, i.e., rules for adjusting model and optimizer hyperparameters (HPs) as model size changes. Some parameterizations fail to transfer optimal base HPs (such as learning rate) across changes in model depth, requiring practitioners to either re-tune these HPs as they scale up (expensive), or accept sub-optimal training when re-tuning is prohibitive. Even when they achieve HP transfer, we develop theory to show parameterizations may still exist in the lazy learning regime where layers learn only features close to their linearization, preventing effective use of depth and nonlinearity. Finally, we identify and adopt the parameterization we call CompleteP that achieves both depth-wise HP transfer and non-lazy learning in all layers. CompleteP enables a wider range of model width/depth ratios to remain compute-efficient, unlocking shapes better suited for different hardware settings and operational contexts. Moreover, CompleteP enables 12-34% compute efficiency improvements over the prior state-of-the-art.

Recent advances in foundation models have emphasized the need to align pre-trained models with specialized domains using small, curated datasets. Studies on these foundation models underscore the importance of low-data training and fine-tuning. This topic, well-known in natural language processing (NLP), has also gained increasing attention in the emerging field of scientific machine learning (SciML). To address the limitations of low-data training and fine-tuning, we draw inspiration from Heavy-Tailed Self-Regularization (HT-SR) theory, analyzing the shape of empirical spectral densities (ESDs) and revealing an imbalance in training quality across different model layers. To mitigate this issue, we adapt a recently proposed layer-wise learning rate scheduler, TempBalance, which effectively balances training quality across layers and enhances low-data training and fine-tuning for both NLP and SciML tasks. Notably, TempBalance demonstrates increasing performance gains as the amount of available tuning data decreases. Comparative analyses further highlight the effectiveness of TempBalance and its adaptability as an "add-on" method for improving model performance.

In class incremental learning, neural networks typically suffer from catastrophic forgetting. We show that an MLP featuring a sparse activation function and an adaptive learning rate optimizer can compete with established regularization techniques in the Split-MNIST task. We highlight the effectiveness of the Adaptive SwisH (ASH) activation function in this context and introduce a novel variant, Hard Adaptive SwisH (Hard ASH) to further enhance the learning retention.

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

Scaling of deep neural networks, especially Transformers, is pivotal for their surging performance and has further led to the emergence of sophisticated reasoning capabilities in foundation models. Such scaling generally requires training large models from scratch with random initialization, failing to leverage the knowledge acquired by their smaller counterparts, which are already resource-intensive to obtain. To tackle this inefficiency, we present LosslEss MOdel ExpansioN (LEMON), a recipe to initialize scaled models using the weights of their smaller but pre-trained counterparts. This is followed by model training with an optimized learning rate scheduler tailored explicitly for the scaled models, substantially reducing the training time compared to training from scratch. Notably, LEMON is versatile, ensuring compatibility with various network structures, including models like Vision Transformers and BERT. Our empirical results demonstrate that LEMON reduces computational costs by 56.7% for Vision Transformers and 33.2% for BERT when compared to training from scratch.

Generative Adversarial Networks (GANs) are a popular formulation to train generative models for complex high dimensional data. The standard method for training GANs involves a gradient descent-ascent (GDA) procedure on a minimax optimization problem. This procedure is hard to analyze in general due to the nonlinear nature of the dynamics. We study the local dynamics of GDA for training a GAN with a kernel-based discriminator. This convergence analysis is based on a linearization of a non-linear dynamical system that describes the GDA iterations, under an isolated points model assumption from [Becker et al. 2022]. Our analysis brings out the effect of the learning rates, regularization, and the bandwidth of the kernel discriminator, on the local convergence rate of GDA. Importantly, we show phase transitions that indicate when the system converges, oscillates, or diverges. We also provide numerical simulations that verify our claims.

The Sharpness Aware Minimization (SAM) optimization algorithm has been shown to control large eigenvalues of the loss Hessian and provide generalization benefits in a variety of settings. The original motivation for SAM was a modified loss function which penalized sharp minima; subsequent analyses have also focused on the behavior near minima. However, our work reveals that SAM provides a strong regularization of the eigenvalues throughout the learning trajectory. We show that in a simplified setting, SAM dynamically induces a stabilization related to the edge of stability (EOS) phenomenon observed in large learning rate gradient descent. Our theory predicts the largest eigenvalue as a function of the learning rate and SAM radius parameters. Finally, we show that practical models can also exhibit this EOS stabilization, and that understanding SAM must account for these dynamics far away from any minima.

Aiming to improve the checkerboard corner detection robustness against the images with poor quality, such as lens distortion, extreme poses, and noise, we propose a novel detection algorithm which can maintain high accuracy on inputs under multiply scenarios without any prior knowledge of the checkerboard pattern. This whole algorithm includes a checkerboard corner detection network and some post-processing techniques. The network model is a fully convolutional network with improvements of loss function and learning rate, which can deal with the images of arbitrary size and produce correspondingly-sized output with a corner score on each pixel by efficient inference and learning. Besides, in order to remove the false positives, we employ three post-processing techniques including threshold related to maximum response, non-maximum suppression, and clustering. Evaluations on two different datasets show its superior robustness, accuracy and wide applicability in quantitative comparisons with the state-of-the-art methods, like MATE, ChESS, ROCHADE and OCamCalib.

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

Previous results have shown that a two time-scale update rule (TTUR) using different learning rates, such as different constant rates or different decaying rates, is useful for training generative adversarial networks (GANs) in theory and in practice. Moreover, not only the learning rate but also the batch size is important for training GANs with TTURs and they both affect the number of steps needed for training. This paper studies the relationship between batch size and the number of steps needed for training GANs with TTURs based on constant learning rates. We theoretically show that, for a TTUR with constant learning rates, the number of steps needed to find stationary points of the loss functions of both the discriminator and generator decreases as the batch size increases and that there exists a critical batch size minimizing the stochastic first-order oracle (SFO) complexity. Then, we use the Fr'echet inception distance (FID) as the performance measure for training and provide numerical results indicating that the number of steps needed to achieve a low FID score decreases as the batch size increases and that the SFO complexity increases once the batch size exceeds the measured critical batch size. Moreover, we show that measured critical batch sizes are close to the sizes estimated from our theoretical results.

Training competitive deep video models is an order of magnitude slower than training their counterpart image models. Slow training causes long research cycles, which hinders progress in video understanding research. Following standard practice for training image models, video model training assumes a fixed mini-batch shape: a specific number of clips, frames, and spatial size. However, what is the optimal shape? High resolution models perform well, but train slowly. Low resolution models train faster, but they are inaccurate. Inspired by multigrid methods in numerical optimization, we propose to use variable mini-batch shapes with different spatial-temporal resolutions that are varied according to a schedule. The different shapes arise from resampling the training data on multiple sampling grids. Training is accelerated by scaling up the mini-batch size and learning rate when shrinking the other dimensions. We empirically demonstrate a general and robust grid schedule that yields a significant out-of-the-box training speedup without a loss in accuracy for different models (I3D, non-local, SlowFast), datasets (Kinetics, Something-Something, Charades), and training settings (with and without pre-training, 128 GPUs or 1 GPU). As an illustrative example, the proposed multigrid method trains a ResNet-50 SlowFast network 4.5x faster (wall-clock time, same hardware) while also improving accuracy (+0.8% absolute) on Kinetics-400 compared to the baseline training method. Code is available online.

A common way to speed up training of large convolutional networks is to add computational units. Training is then performed using data-parallel synchronous Stochastic Gradient Descent (SGD) with mini-batch divided between computational units. With an increase in the number of nodes, the batch size grows. But training with large batch size often results in the lower model accuracy. We argue that the current recipe for large batch training (linear learning rate scaling with warm-up) is not general enough and training may diverge. To overcome this optimization difficulties we propose a new training algorithm based on Layer-wise Adaptive Rate Scaling (LARS). Using LARS, we scaled Alexnet up to a batch size of 8K, and Resnet-50 to a batch size of 32K without loss in accuracy.

Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it notoriously hard to train models with saturating nonlinearities. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer inputs. Our method draws its strength from making normalization a part of the model architecture and performing the normalization for each training mini-batch. Batch Normalization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regularizer, in some cases eliminating the need for Dropout. Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin. Using an ensemble of batch-normalized networks, we improve upon the best published result on ImageNet classification: reaching 4.9% top-5 validation error (and 4.8% test error), exceeding the accuracy of human raters.

Large language models (LLMs) have made remarkable advances in recent years, with scaling laws playing a critical role in this rapid progress. In this paper, we empirically investigate how a critical hyper-parameter, i.e., the global batch size, influences the LLM training prdocess. We begin by training language models ranging from 125 million to 2.6 billion parameters, using up to 300 billion high-quality tokens. Through these experiments, we establish a basic scaling law on model size and training data amount. We then examine how varying batch sizes and learning rates affect the convergence and generalization of these models. Our analysis yields batch size scaling laws under two different cases: with a fixed compute budget, and with a fixed amount of training data. Extrapolation experiments on models of increasing sizes validate our predicted laws, which provides guidance for optimizing LLM training strategies under specific resource constraints.

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation of SGD's failure on Transformers through the lens of Hessian: (i) Transformers are ``heterogeneous'': the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call ``block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs badly on problems with block heterogeneity. To validate that heterogeneity hampers SGD, we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD works well on problems without block heterogeneity but performs badly when the heterogeneity exists. Our initial theoretical analysis indicates that SGD fails because it applies one single learning rate for all blocks, which cannot handle the heterogeneity among blocks. The failure could be rescued if we could assign different learning rates across blocks, as designed in Adam.

Large language models (LLMs) have demonstrated impressive generalization and emergent capabilities, yet their pre-training remains computationally expensive and sensitive to optimization dynamics. While Adam-based optimizers offer fast convergence by adapting learning rates coordinate-wise, recent studies reveal that their updates often suffer from poor spectral conditioning and low-rank structures, hindering efficiency. Muon addresses this issue via global spectral normalization but lacks the per-coordinate adaptivity of Adam. In this work, we propose Column-Normalized Adam (Conda), a novel optimizer that bridges the strengths of both approaches. Conda projects updates into an orthogonal subspace and applies column-wise second moment normalization based on the projected gradients, thereby achieving both improved spectral conditioning and maintaining coordinate-wise adaptivity. This design alleviates the spectral pathologies of Adam while preserving its fast convergence behavior. Extensive experiments on the LLaMA and GPT-2 series show that Conda consistently outperforms AdamW, Muon, and other baselines in pre-training. Remarkably, on the LLaMA series, Conda achieves 2-2.5 the convergence speed of AdamW, measured in both training steps and training time. Further ablations demonstrate its robustness under diverse training setups. These results collectively highlight Conda as an effective and broadly applicable optimizer for large-scale LLM training. The code is released on https://github.com/jie040109/Conda

Training a high-quality deep neural network requires choosing suitable hyperparameters, which is a non-trivial and expensive process. Current works try to automatically optimize or design principles of hyperparameters, such that they can generalize to diverse unseen scenarios. However, most designs or optimization methods are agnostic to the choice of network structures, and thus largely ignore the impact of neural architectures on hyperparameters. In this work, we precisely characterize the dependence of initializations and maximal learning rates on the network architecture, which includes the network depth, width, convolutional kernel size, and connectivity patterns. By pursuing every parameter to be maximally updated with the same mean squared change in pre-activations, we can generalize our initialization and learning rates across MLPs (multi-layer perception) and CNNs (convolutional neural network) with sophisticated graph topologies. We verify our principles with comprehensive experiments. More importantly, our strategy further sheds light on advancing current benchmarks for architecture design. A fair comparison of AutoML algorithms requires accurate network rankings. However, we demonstrate that network rankings can be easily changed by better training networks in benchmarks with our architecture-aware learning rates and initialization.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" [Lewkowycz et al. 2020] that arises when training such models with large learning rates. We then empirically show that the behaviour of neural quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models can be an effective tool for analysis of neural networks.

Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically adjusts the step-size without prior knowledge. To adapt this method to RL settings, we address several issues, including unknown f* in the Polyak step-size. Additionally, we showcase the performance of the Polyak step-size in RL through experiments, demonstrating faster convergence and the attainment of more stable policies.

The rise of large language models (LLMs) has created a significant disparity: industrial research labs with their computational resources, expert teams, and advanced infrastructures, can effectively fine-tune LLMs, while individual developers and small organizations face barriers due to limited resources. In this paper, we aim to bridge this gap by presenting a comprehensive study on supervised fine-tuning of LLMs using instruction-tuning datasets spanning diverse knowledge domains and skills. We focus on small-sized LLMs (3B to 7B parameters) for their cost-efficiency and accessibility. We explore various training configurations and strategies across four open-source pre-trained models. We provide detailed documentation of these configurations, revealing findings that challenge several common training practices, including hyperparameter recommendations from TULU and phased training recommended by Orca. Key insights from our work include: (i) larger batch sizes paired with lower learning rates lead to improved model performance on benchmarks such as MMLU, MTBench, and Open LLM Leaderboard; (ii) early-stage training dynamics, such as lower gradient norms and higher loss values, are strong indicators of better final model performance, enabling early termination of sub-optimal runs and significant computational savings; (iii) through a thorough exploration of hyperparameters like warmup steps and learning rate schedules, we provide guidance for practitioners and find that certain simplifications do not compromise performance; and (iv) we observed no significant difference in performance between phased and stacked training strategies, but stacked training is simpler and more sample efficient. With these findings holding robustly across datasets and models, we hope this study serves as a guide for practitioners fine-tuning small LLMs and promotes a more inclusive environment for LLM research.

Large Language Models (LLM) often needs to be Continual Pre-Trained (CPT) to obtain the unfamiliar language skill or adapt into new domains. The huge training cost of CPT often asks for cautious choice of key hyper-parameters such as the mixture ratio of extra language or domain corpus. However, there is no systematic study which bridge the gap between the optimal mixture ratio and the actual model performance, and the gap between experimental scaling law and the actual deployment in the full model size. In this paper, we perform CPT on Llama-3 8B and 70B to enhance its Chinese ability. We study the optimal correlation between the Additional Language Mixture Ratio (ALMR) and the Learning Rate (LR) on the 8B size which directly indicate the optimal experimental set up. By thorough choice of hyper-parameter, and subsequent fine-tuning, the model capability is improved not only on the Chinese-related benchmark, but also some specific domains including math, coding and emotional intelligence. We deploy the final 70B version of LLM on an real-life chat system which obtain satisfying performance.

With multilingual machine translation (MMT) models continuing to grow in size and number of supported languages, it is natural to reuse and upgrade existing models to save computation as data becomes available in more languages. However, adding new languages requires updating the vocabulary, which complicates the reuse of embeddings. The question of how to reuse existing models while also making architectural changes to provide capacity for both old and new languages has also not been closely studied. In this work, we introduce three techniques that help speed up effective learning of the new languages and alleviate catastrophic forgetting despite vocabulary and architecture mismatches. Our results show that by (1) carefully initializing the network, (2) applying learning rate scaling, and (3) performing data up-sampling, it is possible to exceed the performance of a same-sized baseline model with 30% computation and recover the performance of a larger model trained from scratch with over 50% reduction in computation. Furthermore, our analysis reveals that the introduced techniques help learn the new directions more effectively and alleviate catastrophic forgetting at the same time. We hope our work will guide research into more efficient approaches to growing languages for these MMT models and ultimately maximize the reuse of existing models.

The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam, and (2) accelerated schemes, such as heavy-ball and Nesterov momentum. In this paper, we propose a new optimization algorithm, Lookahead, that is orthogonal to these previous approaches and iteratively updates two sets of weights. Intuitively, the algorithm chooses a search direction by looking ahead at the sequence of fast weights generated by another optimizer. We show that Lookahead improves the learning stability and lowers the variance of its inner optimizer with negligible computation and memory cost. We empirically demonstrate Lookahead can significantly improve the performance of SGD and Adam, even with their default hyperparameter settings on ImageNet, CIFAR-10/100, neural machine translation, and Penn Treebank.

Supervised Fine-Tuning (SFT) on domain-specific datasets is a common approach to adapt Large Language Models (LLMs) to specialized tasks but is often believed to degrade their general capabilities. In this work, we revisit this trade-off and present both empirical and theoretical insights. First, we show that SFT does not always hurt: using a smaller learning rate can substantially mitigate general performance degradation while preserving comparable target-domain performance. We then provide a theoretical analysis that explains these phenomena and further motivates a new method, Token-Adaptive Loss Reweighting (TALR). Building on this, and recognizing that smaller learning rates alone do not fully eliminate general-performance degradation in all cases, we evaluate a range of strategies for reducing general capability loss, including L2 regularization, LoRA, model averaging, FLOW, and our proposed TALR. Experimental results demonstrate that while no method completely eliminates the trade-off, TALR consistently outperforms these baselines in balancing domain-specific gains and general capabilities. Finally, we distill our findings into practical guidelines for adapting LLMs to new domains: (i) using a small learning rate to achieve a favorable trade-off, and (ii) when a stronger balance is further desired, adopt TALR as an effective strategy.

In this paper, we study the role of initialization in Low Rank Adaptation (LoRA) as originally introduced in Hu et al. (2021). Essentially, to start from the pretrained model as initialization for finetuning, one can either initialize B to zero and A to random (default initialization in PEFT package), or vice-versa. In both cases, the product BA is equal to zero at initialization, which makes finetuning starts from the pretrained model. These two initialization schemes are seemingly similar. They should in-principle yield the same performance and share the same optimal learning rate. We demonstrate that this is an incorrect intuition and that the first scheme (initializing B to zero and A to random) on average yields better performance compared to the other scheme. Our theoretical analysis shows that the reason behind this might be that the first initialization allows the use of larger learning rates (without causing output instability) compared to the second initialization, resulting in more efficient learning of the first scheme. We validate our results with extensive experiments on LLMs.

Large artificial neural networks have become a mainstay of language, vision, and audio processing and synthesis, yet their initializations and learning rates are often set in an unsophisticated fashion, due to the high cost of hyperparameter sweeps at scale. The mu-Parameterization (muP) offers a potential solution to this challenge, yielding scaling rules for model initialization and learning rates while reportedly enabling zero-shot hyperparameter transfer from small to large models. Despite its evident promise, the muP method is not yet widely adopted, perhaps due to higher implementation complexity, many variations, or complex theoretical background. This work investigates muP empirically, focusing on the ubiquitous transformer architecture, and aims to answer a simple question: does mu-Transfer yield optimal learning rates in practice? Studying models of up to 10B parameters and training budgets of up to 190B tokens, we find mu-Transfer works as intended for the majority of important cases, yet also identify a few cases where it may not.

We introduce Diff-DOPE, a 6-DoF pose refiner that takes as input an image, a 3D textured model of an object, and an initial pose of the object. The method uses differentiable rendering to update the object pose to minimize the visual error between the image and the projection of the model. We show that this simple, yet effective, idea is able to achieve state-of-the-art results on pose estimation datasets. Our approach is a departure from recent methods in which the pose refiner is a deep neural network trained on a large synthetic dataset to map inputs to refinement steps. Rather, our use of differentiable rendering allows us to avoid training altogether. Our approach performs multiple gradient descent optimizations in parallel with different random learning rates to avoid local minima from symmetric objects, similar appearances, or wrong step size. Various modalities can be used, e.g., RGB, depth, intensity edges, and object segmentation masks. We present experiments examining the effect of various choices, showing that the best results are found when the RGB image is accompanied by an object mask and depth image to guide the optimization process.

As Large Language Models (LLMs) gain widespread practical application, providing the model family of different parameter sizes has become standard practice to address diverse computational requirements. Conventionally, each model in a family is trained independently, resulting in computational costs that scale additively with the number of models. We propose an efficient method for constructing the model family through progressive training, where smaller models are incrementally expanded to larger sizes to create a complete model family. Through extensive experiments with a model family ranging from 1B to 8B parameters, we demonstrate that our method reduces computational costs by approximately 25% while maintaining comparable performance to independently trained models. Furthermore, by strategically adjusting maximum learning rates based on model size, our method outperforms the independent training across various metrics. Beyond performance gains, our approach offers an additional advantage: models in our family tend to yield more consistent behavior across different model sizes.

Merging multiple expert models offers a promising approach for performing multi-task learning without accessing their original data. Existing methods attempt to alleviate task conflicts by sparsifying task vectors or promoting orthogonality among them. However, they overlook the fundamental target of model merging: the merged model performs as closely as possible to task-specific models on respective tasks. We find these methods inevitably discard task-specific information that, while causing conflicts, is crucial for performance. Based on our findings, we frame model merging as a constrained optimization problem (i.e., minimizing the gap between the merged model and individual models, subject to the constraint of retaining shared knowledge) and solve it via adaptive projective gradient descent. Specifically, we align the merged model with individual models by decomposing and reconstituting the loss function, alleviating conflicts through data-free optimization of task vectors. To retain shared knowledge, we optimize this objective by projecting gradients within a shared subspace spanning all tasks. Moreover, we view merging coefficients as adaptive learning rates and propose a task-aware, training-free strategy. Experiments show that our plug-and-play approach consistently outperforms previous methods, achieving state-of-the-art results across diverse architectures and tasks in both vision and NLP domains.

CNNs are increasingly deployed across different hardware, dynamic environments, and low-power embedded devices. This has led to the design and training of CNN architectures with the goal of maximizing accuracy subject to such variable deployment constraints. As the number of deployment scenarios grows, there is a need to find scalable solutions to design and train specialized CNNs. Once-for-all training has emerged as a scalable approach that jointly co-trains many models (subnets) at once with a constant training cost and finds specialized CNNs later. The scalability is achieved by training the full model and simultaneously reducing it to smaller subnets that share model weights (weight-shared shrinking). However, existing once-for-all training approaches incur huge training costs reaching 1200 GPU hours. We argue this is because they either start the process of shrinking the full model too early or too late. Hence, we propose Delayed epsilon-Shrinking (DepsilonpS) that starts the process of shrinking the full model when it is partially trained (~50%) which leads to training cost improvement and better in-place knowledge distillation to smaller models. The proposed approach also consists of novel heuristics that dynamically adjust subnet learning rates incrementally (E), leading to improved weight-shared knowledge distillation from larger to smaller subnets as well. As a result, DEpS outperforms state-of-the-art once-for-all training techniques across different datasets including CIFAR10/100, ImageNet-100, and ImageNet-1k on accuracy and cost. It achieves 1.83% higher ImageNet-1k top1 accuracy or the same accuracy with 1.3x reduction in FLOPs and 2.5x drop in training cost (GPU*hrs)

Dense linear layers are the dominant computational bottleneck in foundation models. Identifying more efficient alternatives to dense matrices has enormous potential for building more compute-efficient models, as exemplified by the success of convolutional networks in the image domain. In this work, we systematically explore structured matrices as replacements for dense matrices. We show that different structures often require drastically different initialization scales and learning rates, which are crucial to performance, especially as models scale. Using insights from the Maximal Update Parameterization, we determine the optimal scaling for initialization and learning rates of these unconventional layers. Finally, we measure the scaling laws of different structures to compare how quickly their performance improves with compute. We propose a novel matrix family containing Monarch matrices, the Block Tensor-Train (BTT), which we show performs better than dense matrices for the same compute on multiple tasks. On CIFAR-10/100 with augmentation, BTT achieves exponentially lower training loss than dense when training MLPs and ViTs. BTT matches dense ViT-S/32 performance on ImageNet-1k with 3.8 times less compute and is more efficient than dense for training small GPT-2 language models.

The dominant line of work in domain adaptation has focused on learning invariant representations using domain-adversarial training. In this paper, we interpret this approach from a game theoretical perspective. Defining optimal solutions in domain-adversarial training as a local Nash equilibrium, we show that gradient descent in domain-adversarial training can violate the asymptotic convergence guarantees of the optimizer, oftentimes hindering the transfer performance. Our analysis leads us to replace gradient descent with high-order ODE solvers (i.e., Runge-Kutta), for which we derive asymptotic convergence guarantees. This family of optimizers is significantly more stable and allows more aggressive learning rates, leading to high performance gains when used as a drop-in replacement over standard optimizers. Our experiments show that in conjunction with state-of-the-art domain-adversarial methods, we achieve up to 3.5% improvement with less than of half training iterations. Our optimizers are easy to implement, free of additional parameters, and can be plugged into any domain-adversarial framework.

Diffusion Transformers have emerged as the foundation for vision generative models, but their scalability is limited by the high cost of hyperparameter (HP) tuning at large scales. Recently, Maximal Update Parametrization (muP) was proposed for vanilla Transformers, which enables stable HP transfer from small to large language models, and dramatically reduces tuning costs. However, it remains unclear whether muP of vanilla Transformers extends to diffusion Transformers, which differ architecturally and objectively. In this work, we generalize standard muP to diffusion Transformers and validate its effectiveness through large-scale experiments. First, we rigorously prove that muP of mainstream diffusion Transformers, including DiT, U-ViT, PixArt-alpha, and MMDiT, aligns with that of the vanilla Transformer, enabling the direct application of existing muP methodologies. Leveraging this result, we systematically demonstrate that DiT-muP enjoys robust HP transferability. Notably, DiT-XL-2-muP with transferred learning rate achieves 2.9 times faster convergence than the original DiT-XL-2. Finally, we validate the effectiveness of muP on text-to-image generation by scaling PixArt-alpha from 0.04B to 0.61B and MMDiT from 0.18B to 18B. In both cases, models under muP outperform their respective baselines while requiring small tuning cost, only 5.5% of one training run for PixArt-alpha and 3% of consumption by human experts for MMDiT-18B. These results establish muP as a principled and efficient framework for scaling diffusion Transformers.

In this paper, we introduce Hunyuan-Large, which is currently the largest open-source Transformer-based mixture of experts model, with a total of 389 billion parameters and 52 billion activation parameters, capable of handling up to 256K tokens. We conduct a thorough evaluation of Hunyuan-Large's superior performance across various benchmarks including language understanding and generation, logical reasoning, mathematical problem-solving, coding, long-context, and aggregated tasks, where it outperforms LLama3.1-70B and exhibits comparable performance when compared to the significantly larger LLama3.1-405B model. Key practice of Hunyuan-Large include large-scale synthetic data that is orders larger than in previous literature, a mixed expert routing strategy, a key-value cache compression technique, and an expert-specific learning rate strategy. Additionally, we also investigate the scaling laws and learning rate schedule of mixture of experts models, providing valuable insights and guidances for future model development and optimization. The code and checkpoints of Hunyuan-Large are released to facilitate future innovations and applications. Codes: https://github.com/Tencent/Hunyuan-Large Models: https://huggingface.co/tencent/Tencent-Hunyuan-Large

In studies of transferable learning, scaling laws are obtained for various important foundation models to predict their properties and performance at larger scales. We show here how scaling law derivation can also be used for model and dataset comparison, allowing to decide which procedure is to be preferred for pre-training. For the first time, full scaling laws based on dense measurements across a wide span of model and samples seen scales are derived for two important language-vision learning procedures, CLIP and MaMMUT, that use either contrastive only or contrastive and captioning text generative loss. Ensuring sufficient prediction accuracy for held out points, we use derived scaling laws to compare both models, obtaining evidence for MaMMUT's stronger improvement with scale and better sample efficiency than standard CLIP. To strengthen validity of the comparison, we show scaling laws for various downstream tasks, classification, retrieval, and segmentation, and for different open datasets, DataComp, DFN and Re-LAION, observing consistently the same trends. We show that comparison can also be performed when deriving scaling laws with a constant learning rate schedule, reducing compute cost. Accurate derivation of scaling laws provides thus means to perform model and dataset comparison across scale spans, avoiding misleading conclusions based on measurements from single reference scales only, paving the road for systematic comparison and improvement of open foundation models and datasets for their creation. We release all the pre-trained models with their intermediate checkpoints, including openMaMMUT-L/14, which achieves 80.3% zero-shot ImageNet-1k accuracy, trained on 12.8B samples from DataComp-1.4B. Code for reproducing experiments in the paper and raw experiments data can be found at https://github.com/LAION-AI/scaling-laws-for-comparison.

A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradients -- gradients through model training -- at scale. We then introduce a "smooth model training" framework that enables effective optimization using metagradients. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.

In recent years, neural models trained on large multilingual text and speech datasets have shown great potential for supporting low-resource languages. This study investigates the performances of two state-of-the-art Automatic Speech Recognition (ASR) models, OpenAI's Whisper (Small & Large-V2) and Facebook's Wav2Vec-BERT on Bangla, a low-resource language. We have conducted experiments using two publicly available datasets: Mozilla Common Voice-17 and OpenSLR to evaluate model performances. Through systematic fine-tuning and hyperparameter optimization, including learning rate, epochs, and model checkpoint selection, we have compared the models based on Word Error Rate (WER), Character Error Rate (CER), Training Time, and Computational Efficiency. The Wav2Vec-BERT model outperformed Whisper across all key evaluation metrics, demonstrated superior performance while requiring fewer computational resources, and offered valuable insights to develop robust speech recognition systems in low-resource linguistic settings.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

This paper demonstrates that a single-layer neural network using Parametric Rectified Linear Unit (PReLU) activation can solve the XOR problem, a simple fact that has been overlooked so far. We compare this solution to the multi-layer perceptron (MLP) and the Growing Cosine Unit (GCU) activation function and explain why PReLU enables this capability. Our results show that the single-layer PReLU network can achieve 100\% success rate in a wider range of learning rates while using only three learnable parameters.

Recently proposed methods for 1-bit and 1.58-bit quantization aware training investigate the performance and behavior of these methods in the context of large language models, finding state-of-the-art performance for models with more than 3B parameters. In this work, we investigate 1.58-bit quantization for small language and vision models ranging from 100K to 48M parameters. We introduce a variant of BitNet b1.58, which allows to rely on the median rather than the mean in the quantization process. Through extensive experiments we investigate the performance of 1.58-bit models obtained through quantization aware training. We further investigate the robustness of 1.58-bit quantization-aware training to changes in the learning rate and regularization through weight decay, finding different patterns for small language and vision models than previously reported for large language models. Our results showcase that 1.58-bit quantization-aware training provides state-of-the-art performance for small language models when doubling hidden layer sizes and reaches or even surpasses state-of-the-art performance for small vision models of identical size. Ultimately, we demonstrate that 1.58-bit quantization-aware training is a viable and promising approach also for training smaller deep learning networks, facilitating deployment of such models in low-resource use-cases and encouraging future research.

Industrial recommender systems face the challenge of operating in non-stationary environments, where data distribution shifts arise from evolving user behaviors over time. To tackle this challenge, a common approach is to periodically re-train or incrementally update deployed deep models with newly observed data, resulting in a continual training process. However, the conventional learning paradigm of neural networks relies on iterative gradient-based updates with a small learning rate, making it slow for large recommendation models to adapt. In this paper, we introduce ReLoop2, a self-correcting learning loop that facilitates fast model adaptation in online recommender systems through responsive error compensation. Inspired by the slow-fast complementary learning system observed in human brains, we propose an error memory module that directly stores error samples from incoming data streams. These stored samples are subsequently leveraged to compensate for model prediction errors during testing, particularly under distribution shifts. The error memory module is designed with fast access capabilities and undergoes continual refreshing with newly observed data samples during the model serving phase to support fast model adaptation. We evaluate the effectiveness of ReLoop2 on three open benchmark datasets as well as a real-world production dataset. The results demonstrate the potential of ReLoop2 in enhancing the responsiveness and adaptiveness of recommender systems operating in non-stationary environments.

Natural language processing is a fast-growing field of artificial intelligence. Since the Transformer was introduced by Google in 2017, a large number of language models such as BERT, GPT, and ELMo have been inspired by this architecture. These models were trained on huge datasets and achieved state-of-the-art results on natural language understanding. However, fine-tuning a pre-trained language model on much smaller datasets for downstream tasks requires a carefully-designed pipeline to mitigate problems of the datasets such as lack of training data and imbalanced data. In this paper, we propose a pipeline to adapt the general-purpose RoBERTa language model to a specific text classification task: Vietnamese Hate Speech Detection. We first tune the PhoBERT on our dataset by re-training the model on the Masked Language Model task; then, we employ its encoder for text classification. In order to preserve pre-trained weights while learning new feature representations, we further utilize different training techniques: layer freezing, block-wise learning rate, and label smoothing. Our experiments proved that our proposed pipeline boosts the performance significantly, achieving a new state-of-the-art on Vietnamese Hate Speech Detection campaign with 0.7221 F1 score.

Gating mechanisms have been widely utilized, from early models like LSTMs and Highway Networks to recent state space models, linear attention, and also softmax attention. Yet, existing literature rarely examines the specific effects of gating. In this work, we conduct comprehensive experiments to systematically investigate gating-augmented softmax attention variants. Specifically, we perform a comprehensive comparison over 30 variants of 15B Mixture-of-Experts (MoE) models and 1.7B dense models trained on a 3.5 trillion token dataset. Our central finding is that a simple modification-applying a head-specific sigmoid gate after the Scaled Dot-Product Attention (SDPA)-consistently improves performance. This modification also enhances training stability, tolerates larger learning rates, and improves scaling properties. By comparing various gating positions and computational variants, we attribute this effectiveness to two key factors: (1) introducing non-linearity upon the low-rank mapping in the softmax attention, and (2) applying query-dependent sparse gating scores to modulate the SDPA output. Notably, we find this sparse gating mechanism mitigates 'attention sink' and enhances long-context extrapolation performance, and we also release related https://github.com/qiuzh20/gated_attention{codes} and https://huggingface.co/QwQZh/gated_attention{models} to facilitate future research.

PaliGemma 2 is an upgrade of the PaliGemma open Vision-Language Model (VLM) based on the Gemma 2 family of language models. We combine the SigLIP-So400m vision encoder that was also used by PaliGemma with the whole range of Gemma 2 models, from the 2B one all the way up to the 27B model. We train these models at three resolutions (224px, 448px, and 896px) in multiple stages to equip them with broad knowledge for transfer via fine-tuning. The resulting family of base models covering different model sizes and resolutions allows us to investigate factors impacting transfer performance (such as learning rate) and to analyze the interplay between the type of task, model size, and resolution. We further increase the number and breadth of transfer tasks beyond the scope of PaliGemma including different OCR-related tasks such as table structure recognition, molecular structure recognition, music score recognition, as well as long fine-grained captioning and radiography report generation, on which PaliGemma 2 obtains state-of-the-art results.

The burgeoning interest in developing Large Language Models (LLMs) with up to trillion parameters has been met with concerns regarding resource efficiency and practical expense, particularly given the immense cost of experimentation. This scenario underscores the importance of exploring the potential of Small Language Models (SLMs) as a resource-efficient alternative. In this context, we introduce MiniCPM, specifically the 1.2B and 2.4B non-embedding parameter variants, not only excel in their respective categories but also demonstrate capabilities on par with 7B-13B LLMs. While focusing on SLMs, our approach exhibits scalability in both model and data dimensions for future LLM research. Regarding model scaling, we employ extensive model wind tunnel experiments for stable and optimal scaling. For data scaling, we introduce a Warmup-Stable-Decay (WSD) learning rate scheduler (LRS), conducive to continuous training and domain adaptation. We present an in-depth analysis of the intriguing training dynamics that occurred in the WSD LRS. With WSD LRS, we are now able to efficiently study data-model scaling law without extensive retraining experiments on both axes of model and data, from which we derive the much higher compute optimal data-model ratio than Chinchilla Optimal. Additionally, we introduce MiniCPM family, including MiniCPM-DPO, MiniCPM-MoE and MiniCPM-128K, whose excellent performance further cementing MiniCPM's foundation in diverse SLM applications. MiniCPM models are available publicly at https://github.com/OpenBMB/MiniCPM .

Empirical scaling laws have driven the evolution of large language models (LLMs), yet their coefficients shift whenever the model architecture or data pipeline changes. Mixture-of-Experts (MoE) models, now standard in state-of-the-art systems, introduce a new sparsity dimension that current dense-model frontiers overlook. We investigate how MoE sparsity influences two distinct capability regimes: memorization and reasoning. We train families of MoE Transformers that systematically vary total parameters, active parameters, and top-k routing while holding the compute budget fixed. For every model we record pre-training loss, downstream task loss, and task accuracy, allowing us to separate the train-test generalization gap from the loss-accuracy gap. Memorization benchmarks improve monotonically with total parameters, mirroring training loss. By contrast, reasoning performance saturates and can even regress despite continued gains in both total parameters and training loss. Altering top-k alone has little effect when active parameters are constant, and classic hyperparameters such as learning rate and initialization modulate the generalization gap in the same direction as sparsity. Neither post-training reinforcement learning (GRPO) nor extra test-time compute rescues the reasoning deficit of overly sparse models. Our model checkpoints, code and logs are open-source at https://github.com/rioyokotalab/optimal-sparsity.

Most public instruction finetuning datasets are relatively small compared to the closed source datasets used to train industry models. To study questions about finetuning at scale, such as curricula and learning rate cooldown schedules, there is a need for industrial-scale datasets. However, this scale necessitates a data generation process that is almost entirely automated. In this work, we study methods for generating large instruction datasets from a single prompt. With little human oversight, we get LLMs to write diverse sets of instruction examples ranging from simple completion tasks to complex multi-turn dialogs across a variety of subject areas. When finetuning a Llama-3 8B base model, our dataset meets or exceeds both WizardLM and Ultrachat on both knowledge-intensive leaderboard tasks as well as conversational evaluations. We release our dataset, the "generator" prompts that created it, and our finetuned model checkpoints.

L_2 regularization and weight decay regularization are equivalent for standard stochastic gradient descent (when rescaled by the learning rate), but as we demonstrate this is not the case for adaptive gradient algorithms, such as Adam. While common implementations of these algorithms employ L_2 regularization (often calling it "weight decay" in what may be misleading due to the inequivalence we expose), we propose a simple modification to recover the original formulation of weight decay regularization by decoupling the weight decay from the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). Our proposed decoupled weight decay has already been adopted by many researchers, and the community has implemented it in TensorFlow and PyTorch; the complete source code for our experiments is available at https://github.com/loshchil/AdamW-and-SGDW

Scalability has driven recent advances in generative modeling, yet its principles remain underexplored for adversarial learning. We investigate the scalability of Generative Adversarial Networks (GANs) through two design choices that have proven to be effective in other types of generative models: training in a compact Variational Autoencoder latent space and adopting purely transformer-based generators and discriminators. Training in latent space enables efficient computation while preserving perceptual fidelity, and this efficiency pairs naturally with plain transformers, whose performance scales with computational budget. Building on these choices, we analyze failure modes that emerge when naively scaling GANs. Specifically, we find issues as underutilization of early layers in the generator and optimization instability as the network scales. Accordingly, we provide simple and scale-friendly solutions as lightweight intermediate supervision and width-aware learning-rate adjustment. Our experiments show that GAT, a purely transformer-based and latent-space GANs, can be easily trained reliably across a wide range of capacities (S through XL). Moreover, GAT-XL/2 achieves state-of-the-art single-step, class-conditional generation performance (FID of 2.96) on ImageNet-256 in just 40 epochs, 6x fewer epochs than strong baselines.

Quantization-aware training (QAT) is a leading technique for improving the accuracy of quantized neural networks. Previous work has shown that decomposing training into a full-precision (FP) phase followed by a QAT phase yields superior accuracy compared to QAT alone. However, the optimal allocation of compute between the FP and QAT phases remains unclear. We conduct extensive experiments with various compute budgets, QAT bit widths, and model sizes from 86.0M to 2.2B to investigate how different QAT durations impact final performance. We demonstrate that, contrary to previous findings, the loss-optimal ratio of QAT to FP training increases with the total amount of compute. Moreover, the optimal fraction can be accurately predicted for a wide range of model sizes and quantization widths using the tokens-per-parameter-byte statistic. From experimental data, we derive a loss scaling law that predicts both optimal QAT ratios and final model performance across different QAT/FP compute allocation strategies and QAT bit widths. We use the scaling law to make further predictions, which we verify experimentally, including which QAT bit width is optimal under a given memory constraint and how QAT accuracy with different bit widths compares to full-precision model accuracy. Additionally, we propose a novel cooldown and QAT fusion approach that performs learning rate decay jointly with quantization-aware training, eliminating redundant full-precision model updates and achieving significant compute savings. These findings provide practical insights into efficient QAT planning and enable the training of higher-quality quantized models with the same compute budget.

Prompt tuning is one of the successful approaches for parameter-efficient tuning of pre-trained language models. Despite being arguably the most parameter-efficient (tuned soft prompts constitute <0.1% of total parameters), it typically performs worse than other efficient tuning methods and is quite sensitive to hyper-parameters. In this work, we introduce Residual Prompt Tuning - a simple and efficient method that significantly improves the performance and stability of prompt tuning. We propose to reparameterize soft prompt embeddings using a shallow network with a residual connection. Our experiments show that Residual Prompt Tuning significantly outperforms prompt tuning on SuperGLUE benchmark. Notably, our method reaches +7 points improvement over prompt tuning with T5-Base and allows to reduce the prompt length by 10x without hurting performance. In addition, we show that our approach is robust to the choice of learning rate and prompt initialization, and is effective in few-shot settings.

In recent years, the application of transformer-based models in time-series forecasting has received significant attention. While often demonstrating promising results, the transformer architecture encounters challenges in fully exploiting the temporal relations within time series data due to its attention mechanism. In this work, we design eXponential Patch (xPatch for short), a novel dual-stream architecture that utilizes exponential decomposition. Inspired by the classical exponential smoothing approaches, xPatch introduces the innovative seasonal-trend exponential decomposition module. Additionally, we propose a dual-flow architecture that consists of an MLP-based linear stream and a CNN-based non-linear stream. This model investigates the benefits of employing patching and channel-independence techniques within a non-transformer model. Finally, we develop a robust arctangent loss function and a sigmoid learning rate adjustment scheme, which prevent overfitting and boost forecasting performance. The code is available at the following repository: https://github.com/stitsyuk/xPatch.

Achieving optimal performance of video diffusion transformers within given data and compute budget is crucial due to their high training costs. This necessitates precisely determining the optimal model size and training hyperparameters before large-scale training. While scaling laws are employed in language models to predict performance, their existence and accurate derivation in visual generation models remain underexplored. In this paper, we systematically analyze scaling laws for video diffusion transformers and confirm their presence. Moreover, we discover that, unlike language models, video diffusion models are more sensitive to learning rate and batch size, two hyperparameters often not precisely modeled. To address this, we propose a new scaling law that predicts optimal hyperparameters for any model size and compute budget. Under these optimal settings, we achieve comparable performance and reduce inference costs by 40.1% compared to conventional scaling methods, within a compute budget of 1e10 TFlops. Furthermore, we establish a more generalized and precise relationship among validation loss, any model size, and compute budget. This enables performance prediction for non-optimal model sizes, which may also be appealed under practical inference cost constraints, achieving a better trade-off.

Kaplan et al. and Hoffmann et al. developed influential scaling laws for the optimal model size as a function of the compute budget, but these laws yield substantially different predictions. We explain the discrepancy by reproducing the Kaplan scaling law on two datasets (OpenWebText2 and RefinedWeb) and identifying three factors causing the difference: last layer computational cost, warmup duration, and scale-dependent optimizer tuning. With these factors corrected, we obtain excellent agreement with the Hoffmann et al. (i.e., "Chinchilla") scaling law. Counter to a hypothesis of Hoffmann et al., we find that careful learning rate decay is not essential for the validity of their scaling law. As a secondary result, we derive scaling laws for the optimal learning rate and batch size, finding that tuning the AdamW beta_2 parameter is essential at lower batch sizes.

We propose a tuning-free dynamic SGD step size formula, which we call Distance over Gradients (DoG). The DoG step sizes depend on simple empirical quantities (distance from the initial point and norms of gradients) and have no ``learning rate'' parameter. Theoretically, we show that a slight variation of the DoG formula enjoys strong parameter-free convergence guarantees for stochastic convex optimization assuming only locally bounded stochastic gradients. Empirically, we consider a broad range of vision and language transfer learning tasks, and show that DoG's performance is close to that of SGD with tuned learning rate. We also propose a per-layer variant of DoG that generally outperforms tuned SGD, approaching the performance of tuned Adam. A PyTorch implementation is available at https://github.com/formll/dog

Adversarial training is exploited to develop a robust Deep Neural Network (DNN) model against the malicious altered data. These attacks may have catastrophic effects on DNN models but are indistinguishable for a human being. For example, an external attack can modify an image adding noises invisible for a human eye, but a DNN model misclassified the image. A key objective for developing robust DNN models is to use a learning algorithm that is fast but can also give model that is robust against different types of adversarial attacks. Especially for adversarial training, enormously long training times are needed for obtaining high accuracy under many different types of adversarial samples generated using different adversarial attack techniques. This paper aims at accelerating the adversarial training to enable fast development of robust DNN models against adversarial attacks. The general method for improving the training performance is the hyperparameters fine-tuning, where the learning rate is one of the most crucial hyperparameters. By modifying its shape (the value over time) and value during the training, we can obtain a model robust to adversarial attacks faster than standard training. First, we conduct experiments on two different datasets (CIFAR10, CIFAR100), exploring various techniques. Then, this analysis is leveraged to develop a novel fast training methodology, AccelAT, which automatically adjusts the learning rate for different epochs based on the accuracy gradient. The experiments show comparable results with the related works, and in several experiments, the adversarial training of DNNs using our AccelAT framework is conducted up to 2 times faster than the existing techniques. Thus, our findings boost the speed of adversarial training in an era in which security and performance are fundamental optimization objectives in DNN-based applications.

Low-Rank Adaptation (LoRA) is a widely used finetuning method for large models. Its small memory footprint allows practitioners to adapt large models to specific tasks at a fraction of the cost of full finetuning. Different modifications have been proposed to enhance its efficiency by, for example, setting the learning rate, the rank, and the initialization. Another improvement axis is adapter placement strategy: when using LoRA, practitioners usually pick module types to adapt with LoRA, such as Query and Key modules. Few works have studied the problem of adapter placement, with nonconclusive results: original LoRA paper suggested placing adapters in attention modules, while other works suggested placing them in the MLP modules. Through an intuitive theoretical analysis, we introduce PLoP (Precise LoRA Placement), a lightweight method that allows automatic identification of module types where LoRA adapters should be placed, given a pretrained model and a finetuning task. We demonstrate that PLoP consistently outperforms, and in the worst case competes, with commonly used placement strategies through comprehensive experiments on supervised finetuning and reinforcement learning for reasoning.

Spectrograms are pivotal in time-frequency signal analysis, widely used in audio processing and computational neuroscience. Chirp-like patterns in electroencephalogram (EEG) spectrograms (marked by linear or exponential frequency sweep) are key biomarkers for seizure dynamics, but automated tools for their detection, localization, and feature extraction are lacking. This study bridges this gap by fine-tuning a Vision Transformer (ViT) model on synthetic spectrograms, augmented with Low-Rank Adaptation (LoRA) to boost adaptability. We generated 100000 synthetic spectrograms with chirp parameters, creating the first large-scale benchmark for chirp localization. These spectrograms mimic neural chirps using linear or exponential frequency sweep, Gaussian noise, and smoothing. A ViT model, adapted for regression, predicted chirp parameters. LoRA fine-tuned the attention layers, enabling efficient updates to the pre-trained backbone. Training used MSE loss and the AdamW optimizer, with a learning rate scheduler and early stopping to curb overfitting. Only three features were targeted: Chirp Start Time (Onset Time), Chirp Start Frequency (Onset Frequency), and Chirp End Frequency (Offset Frequency). Performance was evaluated via Pearson correlation between predicted and actual labels. Results showed strong alignment: 0.9841 correlation for chirp start time, with stable inference times (137 to 140s) and minimal bias in error distributions. This approach offers a tool for chirp analysis in EEG time-frequency representation, filling a critical methodological void.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

Recent advancements in bio-inspired visual sensing and neuromorphic computing have led to the development of various highly efficient bio-inspired solutions with real-world applications. One notable application integrates event-based cameras with spiking neural networks (SNNs) to process event-based sequences that are asynchronous and sparse, making them difficult to handle. In this project, we develop a convolutional spiking neural network (CSNN) architecture that leverages convolutional operations and recurrent properties of a spiking neuron to learn the spatial and temporal relations in the ASL-DVS gesture dataset. The ASL-DVS gesture dataset is a neuromorphic dataset containing hand gestures when displaying 24 letters (A to Y, excluding J and Z due to the nature of their symbols) from the American Sign Language (ASL). We performed classification on a pre-processed subset of the full ASL-DVS dataset to identify letter signs and achieved 100\% training accuracy. Specifically, this was achieved by training in the Google Cloud compute platform while using a learning rate of 0.0005, batch size of 25 (total of 20 batches), 200 iterations, and 10 epochs.

Scaling law principles indicate a power-law correlation between loss and variables such as model size, dataset size, and computational resources utilized during training. These principles play a vital role in optimizing various aspects of model pre-training, ultimately contributing to the success of large language models such as GPT-4, Llama and Gemini. However, the original scaling law paper by OpenAI did not disclose the complete details necessary to derive the precise scaling law formulas, and their conclusions are only based on models containing up to 1.5 billion parameters. Though some subsequent works attempt to unveil these details and scale to larger models, they often neglect the training dependency of important factors such as the learning rate, context length and batch size, leading to their failure to establish a reliable formula for predicting the test loss trajectory. In this technical report, we confirm that the scaling law formulations proposed in the original OpenAI paper remain valid when scaling the model size up to 33 billion, but the constant coefficients in these formulas vary significantly with the experiment setup. We meticulously identify influential factors and provide transparent, step-by-step instructions to estimate all constant terms in scaling-law formulas by training on models with only 1M~60M parameters. Using these estimated formulas, we showcase the capability to accurately predict various attributes for models with up to 33B parameters before their training, including (1) the minimum possible test loss; (2) the minimum required training steps and processed tokens to achieve a specific loss; (3) the critical batch size with an optimal time/computation trade-off at any loss value; and (4) the complete test loss trajectory with arbitrary batch size.

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and f-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-k loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3times faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

Stochastic Gradient Descent (SGD) is one of the many iterative optimization methods that are widely used in solving machine learning problems. These methods display valuable properties and attract researchers and industrial machine learning engineers with their simplicity. However, one of the weaknesses of this type of methods is the necessity to tune learning rate (step-size) for every loss function and dataset combination to solve an optimization problem and get an efficient performance in a given time budget. Stochastic Gradient Descent with Polyak Step-size (SPS) is a method that offers an update rule that alleviates the need of fine-tuning the learning rate of an optimizer. In this paper, we propose an extension of SPS that employs preconditioning techniques, such as Hutchinson's method, Adam, and AdaGrad, to improve its performance on badly scaled and/or ill-conditioned datasets.

Robust Fine-Tuning (RFT) is a low-cost strategy to obtain adversarial robustness in downstream applications, without requiring a lot of computational resources and collecting significant amounts of data. This paper uncovers an issue with the existing RFT, where optimizing both adversarial and natural objectives through the feature extractor (FE) yields significantly divergent gradient directions. This divergence introduces instability in the optimization process, thereby hindering the attainment of adversarial robustness and rendering RFT highly sensitive to hyperparameters. To mitigate this issue, we propose a low-rank (LoRa) branch that disentangles RFT into two distinct components: optimizing natural objectives via the LoRa branch and adversarial objectives via the FE. Besides, we introduce heuristic strategies for automating the scheduling of the learning rate and the scalars of loss terms. Extensive empirical evaluations demonstrate that our proposed automated RFT disentangled via the LoRa branch (AutoLoRa) achieves new state-of-the-art results across a range of downstream tasks. AutoLoRa holds significant practical utility, as it automatically converts a pre-trained FE into an adversarially robust model for downstream tasks without the need for searching hyperparameters.

To advance automated detection of extreme weather events, which are increasing in frequency and intensity with climate change, we explore modifications to a novel light-weight Context Guided convolutional neural network architecture trained for semantic segmentation of tropical cyclones and atmospheric rivers in climate data. Our primary focus is on tropical cyclones, the most destructive weather events, for which current models show limited performance. We investigate feature engineering, data augmentation, learning rate modifications, alternative loss functions, and architectural changes. In contrast to previous approaches optimizing for intersection over union, we specifically seek to improve recall to penalize under-counting and prioritize identification of tropical cyclones. We report success through the use of weighted loss functions to counter class imbalance for these rare events. We conclude with directions for future research on extreme weather events detection, a crucial task for prediction, mitigation, and equitable adaptation to the impacts of climate change.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

We propose NovoGrad, an adaptive stochastic gradient descent method with layer-wise gradient normalization and decoupled weight decay. In our experiments on neural networks for image classification, speech recognition, machine translation, and language modeling, it performs on par or better than well tuned SGD with momentum and Adam or AdamW. Additionally, NovoGrad (1) is robust to the choice of learning rate and weight initialization, (2) works well in a large batch setting, and (3) has two times smaller memory footprint than Adam.

Batch Normalization is a commonly used trick to improve the training of deep neural networks. These neural networks use L2 regularization, also called weight decay, ostensibly to prevent overfitting. However, we show that L2 regularization has no regularizing effect when combined with normalization. Instead, regularization has an influence on the scale of weights, and thereby on the effective learning rate. We investigate this dependence, both in theory, and experimentally. We show that popular optimization methods such as ADAM only partially eliminate the influence of normalization on the learning rate. This leads to a discussion on other ways to mitigate this issue.

We present RWKV-7 "Goose", a new sequence modeling architecture, along with pre-trained language models that establish a new state-of-the-art in downstream performance at the 3 billion parameter scale on multilingual tasks, and match current SoTA English language performance despite being trained on dramatically fewer tokens than other top 3B models. Nevertheless, RWKV-7 models require only constant memory usage and constant inference time per token. RWKV-7 introduces a newly generalized formulation of the delta rule with vector-valued gating and in-context learning rates, as well as a relaxed value replacement rule. We show that RWKV-7 can perform state tracking and recognize all regular languages, while retaining parallelizability of training. This exceeds the capabilities of Transformers under standard complexity conjectures, which are limited to TC^0. To demonstrate RWKV-7's language modeling capability, we also present an extended open source 3.1 trillion token multilingual corpus, and train four RWKV-7 models ranging from 0.19 billion to 2.9 billion parameters on this dataset. To foster openness, reproduction, and adoption, we release our models and dataset component listing at https://huggingface.co/RWKV, and our training and inference code at https://github.com/RWKV/RWKV-LM all under the Apache 2.0 License.

While Multimodal Large Language Models (MLLMs) have achieved impressive progress in vision-language understanding, they still struggle with complex multi-step reasoning, often producing logically inconsistent or partially correct solutions. A key limitation lies in the lack of fine-grained supervision over intermediate reasoning steps. To address this, we propose MM-PRM, a process reward model trained within a fully automated, scalable framework. We first build MM-Policy, a strong multimodal model trained on diverse mathematical reasoning data. Then, we construct MM-K12, a curated dataset of 10,000 multimodal math problems with verifiable answers, which serves as seed data. Leveraging a Monte Carlo Tree Search (MCTS)-based pipeline, we generate over 700k step-level annotations without human labeling. The resulting PRM is used to score candidate reasoning paths in the Best-of-N inference setup and achieves significant improvements across both in-domain (MM-K12 test set) and out-of-domain (OlympiadBench, MathVista, etc.) benchmarks. Further analysis confirms the effectiveness of soft labels, smaller learning rates, and path diversity in optimizing PRM performance. MM-PRM demonstrates that process supervision is a powerful tool for enhancing the logical robustness of multimodal reasoning systems. We release all our codes and data at https://github.com/ModalMinds/MM-PRM.

Recent years have seen a growing interest and adoption of LLMs, with muTransfer becoming a key technique for tuning hyperparameters in large-scale training. Meanwhile, Mixture-of-Experts (MoE) has emerged as a leading architecture in extremely large models. However, the intersection of these two advancements has remained unexplored. In this work, we derive a mu-Parameterization (muP) for MoE, providing theoretical guarantees for feature learning across model widths in both the router and experts. We empirically validate our parameterization and further investigate how scaling the number of experts and granularity affects the optimal learning rate.

Disease progression simulation is a crucial area of research that has significant implications for clinical diagnosis, prognosis, and treatment. One major challenge in this field is the lack of continuous medical imaging monitoring of individual patients over time. To address this issue, we develop a novel framework termed Progressive Image Editing (PIE) that enables controlled manipulation of disease-related image features, facilitating precise and realistic disease progression simulation. Specifically, we leverage recent advancements in text-to-image generative models to simulate disease progression accurately and personalize it for each patient. We theoretically analyze the iterative refining process in our framework as a gradient descent with an exponentially decayed learning rate. To validate our framework, we conduct experiments in three medical imaging domains. Our results demonstrate the superiority of PIE over existing methods such as Stable Diffusion Walk and Style-Based Manifold Extrapolation based on CLIP score (Realism) and Disease Classification Confidence (Alignment). Our user study collected feedback from 35 veteran physicians to assess the generated progressions. Remarkably, 76.2% of the feedback agrees with the fidelity of the generated progressions. To our best knowledge, PIE is the first of its kind to generate disease progression images meeting real-world standards. It is a promising tool for medical research and clinical practice, potentially allowing healthcare providers to model disease trajectories over time, predict future treatment responses, and improve patient outcomes.

Recent advances in table understanding have focused on instruction-tuning large language models (LLMs) for table-related tasks. However, existing research has overlooked the impact of hyperparameter choices and lacks a comprehensive evaluation of the out-of-domain table understanding ability and the general capabilities of these table LLMs. In this paper, we evaluate these abilities in existing table LLMs, and reveal significant declines in both out-of-domain table understanding and general capabilities compared to their base models. Through systematic analysis, we show that hyperparameters, such as learning rate, can significantly influence both table-specific and general capabilities. Contrary to the existing table instruction-tuning works, we demonstrate that smaller learning rates and fewer training instances can enhance table understanding while preserving general capabilities. Based on our findings, we introduce TAMA, a TAble LLM instruction-tuned from LLaMA 3.1 8B Instruct, which achieves performance on par with, or surpassing GPT-3.5 and GPT-4 on table tasks, while maintaining strong out-of-domain generalization and general capabilities. Our findings highlight the potential for reduced data annotation costs and more efficient model development through careful hyperparameter selection.

The massive computational costs associated with large language model (LLM) pretraining have spurred great interest in reduced-precision floating-point representations to accelerate the process. As a result, the BrainFloat16 (BF16) precision has become the de facto standard for LLM training, with hardware support included in recent accelerators. This trend has gone even further in the latest processors, where FP8 has recently been introduced. However, prior experience with FP16, which was found to be less stable than BF16, raises concerns as to whether FP8, with even fewer bits than FP16, can be a cost-effective option for LLM training. We argue that reduced-precision training schemes must have similar training stability and hyperparameter sensitivities to their higher-precision counterparts in order to be cost-effective. However, we find that currently available methods for FP8 training are not robust enough to allow their use as economical replacements. This prompts us to investigate the stability of reduced-precision LLM training in terms of robustness across random seeds and learning rates. To this end, we propose new evaluation techniques and a new metric for quantifying loss landscape sharpness in autoregressive language models. By simulating incremental bit reductions in floating-point representations, we analyze the relationship between representational power and training stability with the intent of aiding future research into the field.

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an epsilon-stationary point with an iteration complexity of mathcal O(epsilon^{-4}). Our complexity results for both RMSProp and Adam match with the complexity lower bound established in arjevani2023lower.

In this report, we describe the technical details of our submission to the 2023 ABO Fine-grained Semantic Segmentation Competition, by Team "Zeyu\_Dong" (username:ZeyuDong). The task is to predicate the semantic labels for the convex shape of five categories, which consist of high-quality, standardized 3D models of real products available for purchase online. By using DGCNN as the backbone to classify different structures of five classes, We carried out numerous experiments and found learning rate stochastic gradient descent with warm restarts and setting different rate of factors for various categories contribute most to the performance of the model. The appropriate method helps us rank 3rd place in the Dev phase of the 2023 ICCV 3DVeComm Workshop Challenge.

Most real-world 3D measurements from depth sensors are incomplete, and to address this issue the point cloud completion task aims to predict the complete shapes of objects from partial observations. Previous works often adapt an encoder-decoder architecture, where the encoder is trained to extract embeddings that are used as inputs to generate predictions from the decoder. However, the learned embeddings have sparse distribution in the feature space, which leads to worse generalization results during testing. To address these problems, this paper proposes a hyperspherical module, which transforms and normalizes embeddings from the encoder to be on a unit hypersphere. With the proposed module, the magnitude and direction of the output hyperspherical embedding are decoupled and only the directional information is optimized. We theoretically analyze the hyperspherical embedding and show that it enables more stable training with a wider range of learning rates and more compact embedding distributions. Experiment results show consistent improvement of point cloud completion in both single-task and multi-task learning, which demonstrates the effectiveness of the proposed method.

We introduce Bielik 7B v0.1, a 7-billion-parameter generative text model for Polish language processing. Trained on curated Polish corpora, this model addresses key challenges in language model development through innovative techniques. These include Weighted Instruction Cross-Entropy Loss, which balances the learning of different instruction types, and Adaptive Learning Rate, which dynamically adjusts the learning rate based on training progress. To evaluate performance, we created the Open PL LLM Leaderboard and Polish MT-Bench, novel frameworks assessing various NLP tasks and conversational abilities. Bielik 7B v0.1 demonstrates significant improvements, achieving a 9 percentage point increase in average score compared to Mistral-7B-v0.1 on the RAG Reader task. It also excels in the Polish MT-Bench, particularly in Reasoning (6.15/10) and Role-playing (7.83/10) categories. This model represents a substantial advancement in Polish language AI, offering a powerful tool for diverse linguistic applications and setting new benchmarks in the field.