Optimism in Reinforcement Learning and Kullback-Leibler Divergence

TL;DR

This paper advocates for using Kullback-Leibler divergence in optimistic model-based reinforcement learning, introducing KL-UCRL, which matches UCRL2's theoretical guarantees but shows improved empirical performance, especially in less connected MDPs.

Contribution

The paper introduces KL-UCRL, an efficient algorithm leveraging KL divergence for optimism in reinforcement learning, with theoretical regret guarantees and improved empirical results.

Findings

KL-UCRL matches UCRL2's regret bounds.

Numerical experiments show better performance in sparse MDPs.

Geometric analysis explains the improved behavior.

Abstract

We consider model-based reinforcement learning in finite Markov De- cision Processes (MDPs), focussing on so-called optimistic strategies. In MDPs, optimism can be implemented by carrying out extended value it- erations under a constraint of consistency with the estimated model tran- sition probabilities. The UCRL2 algorithm by Auer, Jaksch and Ortner (2009), which follows this strategy, has recently been shown to guarantee near-optimal regret bounds. In this paper, we strongly argue in favor of using the Kullback-Leibler (KL) divergence for this purpose. By studying the linear maximization problem under KL constraints, we provide an ef- ficient algorithm, termed KL-UCRL, for solving KL-optimistic extended value iteration. Using recent deviation bounds on the KL divergence, we prove that KL-UCRL provides the same guarantees as UCRL2 in terms of regret. However, numerical experiments on classical benchmarks show a significantly improved behavior, particularly when the MDP has reduced connectivity. To support this observation, we provide elements of com- parison between the two algorithms based on geometric considerations.