Sharp Dichotomies for Regret Minimization in Metric Spaces

TL;DR

This paper establishes a universal dichotomy in regret bounds for Lipschitz multi-armed bandit problems across all metric spaces, linking the bounds to topological properties like compactness and countability.

Contribution

It proves a universal dichotomy for regret bounds in Lipschitz MAB problems, connecting online learning bounds with classical topology.

Findings

Regret is either logarithmic or sublinear, depending on the metric space's topology.

The dichotomy depends on whether the metric space's completion is compact and countable.

A similar dichotomy is shown for the full-feedback (best-expert) setting.

Abstract

The Lipschitz multi-armed bandit (MAB) problem generalizes the classical multi-armed bandit problem by assuming one is given side information consisting of a priori upper bounds on the difference in expected payoff between certain pairs of strategies. Classical results of (Lai and Robbins 1985) and (Auer et al. 2002) imply a logarithmic regret bound for the Lipschitz MAB problem on finite metric spaces. Recent results on continuum-armed bandit problems and their generalizations imply lower bounds of $\sqrt{t}$, or stronger, for many infinite metric spaces such as the unit interval. Is this dichotomy universal? We prove that the answer is yes: for every metric space, the optimal regret of a Lipschitz MAB algorithm is either bounded above by any $f\in \omega(\log t)$, or bounded below by any $g\in o(\sqrt{t})$. Perhaps surprisingly, this dichotomy does not coincide with the distinction between finite and infinite metric spaces; instead it depends on whether the completion of the metric space is compact and countable. Our proof connects upper and lower bound techniques in online learning with classical topological notions such as perfect sets and the Cantor-Bendixson theorem. Among many other results, we show a similar dichotomy for the "full-feedback" (a.k.a., "best-expert") version.