Boltzmann Exploration Done Right

TL;DR

This paper critically examines Boltzmann exploration in multi-armed bandits, revealing its limitations with monotone learning rates and proposing novel non-monotone and arm-specific strategies that achieve near-optimal regret bounds.

Contribution

It introduces a non-monotone schedule and a new arm-specific learning rate variant that improve exploration efficiency and regret bounds without prior problem knowledge.

Findings

Monotone Boltzmann exploration induces suboptimal behavior.

A non-monotone schedule can achieve near-optimal performance with prior knowledge.

Arm-specific learning rates attain optimal regret bounds without prior knowledge.

Abstract

Boltzmann exploration is a classic strategy for sequential decision-making under uncertainty, and is one of the most standard tools in Reinforcement Learning (RL). Despite its widespread use, there is virtually no theoretical understanding about the limitations or the actual benefits of this exploration scheme. Does it drive exploration in a meaningful way? Is it prone to misidentifying the optimal actions or spending too much time exploring the suboptimal ones? What is the right tuning for the learning rate? In this paper, we address several of these questions in the classic setup of stochastic multi-armed bandits. One of our main results is showing that the Boltzmann exploration strategy with any monotone learning-rate sequence will induce suboptimal behavior. As a remedy, we offer a simple non-monotone schedule that guarantees near-optimal performance, albeit only when given prior access to key problem parameters that are typically not available in practical situations (like the time horizon $T$ and the suboptimality gap $\Delta$). More importantly, we propose a novel variant that uses different learning rates for different arms, and achieves a distribution-dependent regret bound of order $\frac{K\log^2 T}{\Delta}$ and a distribution-independent bound of order $\sqrt{KT}\log K$ without requiring such prior knowledge. To demonstrate the flexibility of our technique, we also propose a variant that guarantees the same performance bounds even if the rewards are heavy-tailed.