We propose a general framework to design posterior sampling methods for model-based RL. We show that the proposed algorithms can be analyzed by reducing regret to Hellinger distance in conditional probability estimation. We further show that optimistic posterior sampling can control this Hellinger distance, when we measure model error via data likelihood. This technique allows us to design and analyze unified posterior sampling algorithms with state-of-the-art sample complexity guarantees for many model-based RL settings. We illustrate our general result in many special cases, demonstrating the versatility of our framework.

We consider reinforcement learning in an environment modeled by an episodic, tabular, step-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 $O(\sqrt{H^3SAT})$ ignoring $\text{poly}\log(HSAT)$ terms. The key novel technical ingredient is a new sharp anti-concentration inequality for linear forms of a Dirichlet random vector 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 (2017) for the episodic setting.

We propose a general framework to design posterior sampling methods for model-based RL. We show that the proposed algorithms can be analyzed by reducing regret to Hellinger distance in conditional probability estimation. We further show that optimistic posterior sampling can control this Hellinger distance, when we measure model error via data likelihood. This technique allows us to design and analyze unified posterior sampling algorithms with state-of-the-art sample complexity guarantees for many model-based RL settings. We illustrate our general result in many special cases, demonstrating the versatility of our framework.

We consider reinforcement learning in an environment modeled by an episodic, tabular, step-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 3SAT}) ignoring \text{poly}\log(HSAT) terms. The key novel technical ingredient is a new sharp anti-concentration inequality for linear forms of a Dirichlet random vector which may be of independent interest.