Minimal Exploration in Structured Stochastic Bandits

TL;DR

This paper develops a unified framework for structured stochastic bandit problems, deriving a fundamental regret lower bound and introducing OSSB, an algorithm that asymptotically matches this limit without relying on classical exploration strategies.

Contribution

It provides the first instance-specific regret lower bound for a broad class of structured bandits and introduces OSSB, a novel algorithm that achieves this bound.

Findings

OSSB outperforms existing algorithms like Thompson sampling in linear bandits.

The regret lower bound is asymptotically tight and instance-specific.

Numerical experiments validate OSSB's efficiency across various structures.

Abstract

This paper introduces and addresses a wide class of stochastic bandit problems where the function mapping the arm to the corresponding reward exhibits some known structural properties. Most existing structures (e.g. linear, Lipschitz, unimodal, combinatorial, dueling, ...) are covered by our framework. We derive an asymptotic instance-specific regret lower bound for these problems, and develop OSSB, an algorithm whose regret matches this fundamental limit. OSSB is not based on the classical principle of "optimism in the face of uncertainty" or on Thompson sampling, and rather aims at matching the minimal exploration rates of sub-optimal arms as characterized in the derivation of the regret lower bound. We illustrate the efficiency of OSSB using numerical experiments in the case of the linear bandit problem and show that OSSB outperforms existing algorithms, including Thompson sampling.