Bandit Regret Scaling with the Effective Loss Range

TL;DR

This paper explores how the regret bounds of nonstochastic multi-armed bandits can be improved when the effective loss range is small, introducing new assumptions and techniques to achieve better guarantees.

Contribution

It introduces a novel method to convert bandit algorithms with loss range dependence into ones depending only on the effective range, under mild assumptions.

Findings

Regret bounds can be improved with small effective loss range

New technique converts loss range-dependent algorithms to effective range-dependent

Achieves better regret guarantees under mild additional assumptions

Abstract

We study how the regret guarantees of nonstochastic multi-armed bandits can be improved, if the effective range of the losses in each round is small (e.g. the maximal difference between two losses in a given round). Despite a recent impossibility result, we show how this can be made possible under certain mild additional assumptions, such as availability of rough estimates of the losses, or advance knowledge of the loss of a single, possibly unspecified arm. Along the way, we develop a novel technique which might be of independent interest, to convert any multi-armed bandit algorithm with regret depending on the loss range, to an algorithm with regret depending only on the effective range, while avoiding predictably bad arms altogether.