Refined Regret for Adversarial MDPs with Linear Function Approximation

TL;DR

This paper introduces two algorithms that significantly improve regret bounds for adversarial MDPs with linear function approximation, achieving near-optimal regret in various settings and extending to simulator-free cases.

Contribution

It presents two novel algorithms with refined analysis and loss estimators that improve regret bounds for adversarial MDPs with linear function approximation, including simulator-free scenarios.

Findings

Achieves $ ilde{O}( oot K)$ regret with the first algorithm.

Develops a magnitude-reduced loss estimator for better regret bounds.

Extends to simulator-free linear MDPs with improved regret bounds.

Abstract

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over $K$ episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order $\tilde{\mathcal O}(K^{2/3})$ (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to $\tilde{\mathcal O}(\sqrt K)$ in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves $\tilde{\mathcal O}(K^{8/9})$ regret and greatly improves over the best existing bound $\tilde{\mathcal O}(K^{14/15})$. This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.