Lipschitz Bandits: Regret Lower Bounds and Optimal Algorithms

TL;DR

This paper establishes regret lower bounds and introduces optimal algorithms for Lipschitz bandit problems, demonstrating their effectiveness in both discrete and continuous settings through theoretical analysis and numerical experiments.

Contribution

It provides the first asymptotic regret lower bounds for Lipschitz bandits and proposes algorithms that are proven to be asymptotically optimal, extending to continuous and contextual cases.

Findings

OSLB is asymptotically optimal for discrete Lipschitz bandits.

Discretization combined with OSLB or CKL-UCB outperforms existing methods.

Algorithms extend effectively to contextual bandits with similarities.

Abstract

We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for weighted sums of KL divergences between the empirical distributions of rewards and their true distributions. For continuous Lipschitz bandits, we propose to first discretize the action space, and then apply OSLB or CKL-UCB, algorithms that provably exploit the structure efficiently. This approach is shown, through numerical experiments, to significantly outperform existing algorithms that directly deal with the continuous set of arms. Finally the results and algorithms are extended to contextual bandits with similarities.