Optimistic posterior sampling for reinforcement learning with few samples and tight guarantees

Daniil Tiapkin, Denis Belomestny, Daniele Calandriello, Éric Moulines, Rémi Munos, Alexey Naumov, Mark Rowland, Michal Valko, Pierre Ménard

2022 · in (NeurIPS 2022)

Abstract

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 O(sqrt(H^3 SAT)) ignoring 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 Omega(sqrt(H^3 SAT)), thereby answering the open problems raised by Agrawal and Jia (2017b) for the episodic setting.

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