Reward-oriented Causal Representation Learning
Zirui Yan, Emre Acartürk, Ali Tajer
Causal representation learning (CRL) is the process of disentangling the **latent** low-dimensional causally-related generating factors underlying high-dimensional observable data. Extensive recent studies have characterized CRL identifiability and **perfect** recovery of the latent variables and their attendant causal graph. This paper introduces the notion of **reward-oriented** CRL, the purpose of which is moving away from perfectly learning the latent representation and instead learning it the extended needed for optimizing a desired downstream task (reward). In reward-oriented CRL, perfectly learning the latent representation can be excessive, and instead it has to be learned at the **coarsest** level that is sufficient for optimizing the desired task. Reward-oriented CRL is formalized as optimizing a desired function of the observable data over the space of all possible interventions and focuses on linear causal and transformation models. With the objective of sequentially identifying the optimal subset of interventions, an algorithm with adaptive exploration is designed that learns the causal graph $\mathcal{G}$ and variables needed to identify the best intervention. It is shown that for an $n$-dimensional latent space and a $d$-dimensional observation space, over a horizon $T$ the regret upper bound scales as $\tilde O(d^{\frac{1}{2}}n^{\frac{1}{3}}u^{\frac{2}{3}}T^{\frac{2}{3}} + u\sqrt{T})$. Furthermore, an almost-matching lower bound is shown to scale as $\Omega(d^{\frac{1}{2}}n^{\frac{1}{3}}p^{\frac{2}{3}}T^{\frac{2}{3}} + p\sqrt{T})$, where $u$ measures total uncertainty in the graph estimates and $p$ counts the number of causal paths in the graph.