Regularization and Variance-Weighted Regression Achieves Minimax Optimality in Linear MDPs: Theory and Practice
This work addresses the problem of sample efficiency in reinforcement learning for researchers and practitioners, offering both theoretical optimality and practical gains, though it is incremental in extending MDVI to linear function approximation.
The paper tackles the theoretical gap in mirror descent value iteration (MDVI) with linear function approximation by introducing Variance-Weighted Least-Squares MDVI (VWLS-MDVI), which achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs, and proposes a practical deep RL algorithm, Deep Variance Weighting (DVW), that improves performance on MinAtar benchmarks.
Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an $\varepsilon$-optimal policy with probability $1-δ$ under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.