Reinforcement Learning for Markovian Bandits: Is Posterior Sampling more Scalable than Optimism?

TL;DR

This paper introduces MB-PSRL and MB-UCRL2 algorithms for Markovian bandits, demonstrating their near-optimal regret bounds, computational efficiency, and practical superiority over classical methods.

Contribution

The paper adapts PSRL and UCRL2 to Markovian bandits, achieving scalable algorithms with near-optimal regret and linear runtime in the number of bandits.

Findings

MB-PSRL and MB-UCRL2 have regret bounds of O(Ssqrt{nK})

MB-PSRL runtime is linear in the number of bandits

Numerical experiments show MB-PSRL outperforms existing algorithms

Abstract

We study learning algorithms for the classical Markovian bandit problem with discount. We explain how to adapt PSRL [24] and UCRL2 [2] to exploit the problem structure. These variants are called MB-PSRL and MB-UCRL2. While the regret bound and runtime of vanilla implementations of PSRL and UCRL2 are exponential in the number of bandits, we show that the episodic regret of MB-PSRL and MB-UCRL2 is $\tilde{O}(S\sqrt{nK})$ where $K$ is the number of episodes, $n$ is the number of bandits and $S$ is the number of states of each bandit (the exact bound in S, n and K is given in the paper). Up to a factor $\sqrt S$, this matches the lower bound of $\Omega(\sqrt{SnK})$ that we also derive in the paper. MB-PSRL is also computationally efficient: its runtime is linear in the number of bandits. We further show that this linear runtime cannot be achieved by adapting classical non-Bayesian algorithms such as UCRL2 or UCBVI to Markovian bandit problems. Finally, we perform numerical experiments that confirm that MB-PSRL outperforms other existing algorithms in practice, both in terms of regret and of computation time.