Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees

• URL: http://arxiv.org/abs/2209.14414v1
• Date: Wed, 28 Sep 2022 20:49:34 GMT
• Title: Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees
• Authors: Daniil Tiapkin, Denis Belomestny, Daniele Calandriello, Eric Moulines, Remi Munos, Alexey Naumov, Mark Rowland, Michal Valko, Pierre Menard
• Abstract summary: We propose an optimistic posterior sampling algorithm for reinforcement learning (OPSRL) We guarantee a high-probability regret bound of order at most $widetildemathcalO(sqrtH3SAT)$ ignoring $textpolylog(HSAT)$ terms. Our bound matches the lower bound of order $Omega(sqrtH3SAT)$, thereby answering the open problems raised by Agrawal and Jia.
• Score: 43.13918072870693
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ 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 $\Omega(\sqrt{H^3SAT})$, thereby answering the open problems raised by Agrawal and Jia [2017b] for the episodic setting.

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