You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
Deep neural networks with remarkably strong generalization performances are usually over-parameterized. Despite explicit regularization strategies are used for practitioners to avoid over-fitting, the impacts are often small. Some theoretical studies have analyzed the implicit regularization effect of stochastic gradient descent (SGD) on simple machine learning models with certain assumptions. However, how it behaves practically in state-of-the-art models and real-world datasets is still unknown. To bridge this gap, we study the role of SGD implicit regularization in deep learning systems. We show pure SGD tends to converge to minimas that have better generalization performances in multiple natural language processing (NLP) tasks. This phenomenon coexists with dropout, an explicit regularizer. In addition, neural network's finite learning capability does not impact the intrinsic nature of SGD's implicit regularization effect. Specifically, under limited training samples or with certain corrupted labels, the implicit regularization effect remains strong. We further analyze the stability by varying the weight initialization range. We corroborate these experimental findings with a decision boundary visualization using a 3-layer neural network for interpretation. Altogether, our work enables a deepened understanding on how implicit regularization affects the deep learning model and sheds light on the future study of the over-parameterized model's generalization ability.

Cited Abstract(s):
[1]: We show that for a general class of convex online learning problems, Mirror Descent can always achieve a (nearly) optimal regret guarantee.
A few papers have constructed theories of SGD implicit regularization. li2017algorithmic prove the implicit regularization effect of SGD on small machine learning models based on the fact that SGD prefers some generalizable local minimas to others [1] . However, these theories does not establish universal conclusions which can be applied to deeper neural networks with more practical activations and the cross-entropy loss function. Such theory construction requires a better understanding to optimization algorithm's behavior for non-linear models and statistics. In this work, we study the unsolved problem base on large-scale and comprehensive empirical studies.