Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees
This work provides a theoretically optimal algorithm for reinforcement learning with tight sample efficiency guarantees, solving a foundational open problem for researchers in theoretical RL.
The paper tackles the problem of reinforcement learning in episodic Markov decision processes by proposing an optimistic posterior sampling algorithm (OPSRL) that achieves a high-probability regret bound of order at most ̃O(√(H^3SAT)), matching the lower bound and addressing open problems in the field.
We consider reinforcement learning in an environment modeled by an episodic, finite, stage-dependent Markov decision process of horizon $H$ with $S$ states, and $A$ actions. The performance of an agent is measured by the regret after interacting with the environment for $T$ episodes. We propose an optimistic posterior sampling algorithm for reinforcement learning (OPSRL), a simple variant of posterior sampling that only needs a number of posterior samples logarithmic in $H$, $S$, $A$, and $T$ per state-action pair. For OPSRL we guarantee a high-probability regret bound of order at most $\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ ignoring $\text{poly}\log(HSAT)$ terms. The key novel technical ingredient is a new sharp anti-concentration inequality for linear forms which may be of independent interest. Specifically, we extend the normal approximation-based lower bound for Beta distributions by Alfers and Dinges [1984] to Dirichlet distributions. Our bound matches the lower bound of order $Ω(\sqrt{H^3SAT})$, thereby answering the open problems raised by Agrawal and Jia [2017b] for the episodic setting.