Lipschitz Bandits without the Lipschitz Constant

TL;DR

This paper develops a bandit strategy that adapts to the environment's Lipschitz properties without prior knowledge of the Lipschitz constant or time horizon, achieving optimal regret bounds.

Contribution

It introduces a method for Lipschitz bandits that attains minimax optimal regret without needing to know the Lipschitz constant or time horizon beforehand.

Findings

Achieves optimal regret bounds without prior knowledge of L or T.

Provides a strategy that adapts to the environment's Lipschitz properties.

Outperforms previous methods requiring known Lipschitz constants.

Abstract

We consider the setting of stochastic bandit problems with a continuum of arms. We first point out that the strategies considered so far in the literature only provided theoretical guarantees of the form: given some tuning parameters, the regret is small with respect to a class of environments that depends on these parameters. This is however not the right perspective, as it is the strategy that should adapt to the specific bandit environment at hand, and not the other way round. Put differently, an adaptation issue is raised. We solve it for the special case of environments whose mean-payoff functions are globally Lipschitz. More precisely, we show that the minimax optimal orders of magnitude $L^{d/(d+2)} \, T^{(d+1)/(d+2)}$ of the regret bound against an environment $f$ with Lipschitz constant $L$ over $T$ time instances can be achieved without knowing $L$ or $T$ in advance. This is in contrast to all previously known strategies, which require to some extent the knowledge of $L$ to achieve this performance guarantee.