Title: Boltzmann Exploration Done Right,
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.