Computationally Efficient Horizon-Free Reinforcement Learning for Linear Mixture MDPs

Abstract

Recent studies have shown that episodic reinforcement learning (RL) is not more difficult than contextual bandits, even with a long planning horizon and unknown state transitions. However, these results are limited to either tabular Markov decision processes (MDPs) or computationally inefficient algorithms for linear mixture MDPs. In this paper, we propose the first computationally efficient horizon-free algorithm for linear mixture MDPs, which achieves the optimal $\tilde{O}(d\sqrt{K}+d^2)$ regret up to logarithmic factors. Our algorithm adapts a weighted least square estimator for the unknown transitional dynamic, where the weight is both \emph{variance-aware} and \emph{uncertainty-aware}. When applying our weighted least square estimator to heterogeneous linear bandits, we can obtain an $\tilde O(d\sqrt{\sum_{k=1}^K \sigma_k^2} +d)$regretinthefirst$K$rounds,where$d$ is the dimension of the context and ${\sigma }_k^2$ is the variance of the reward in the $k$-th round. This also improves upon the best-known algorithms in this setting when ${\sigma }_k^2$'s are known.