More Adaptive Algorithms for Adversarial Bandits

TL;DR

This paper introduces a versatile, parameter-free algorithm for adversarial multi-armed bandits that achieves various data-dependent regret bounds, improving upon previous methods and adapting to different problem complexities.

Contribution

The paper presents a new adaptive algorithm based on Online Mirror Descent with a log-barrier regularizer, achieving multiple novel regret bounds and enhanced adaptability.

Findings

Achieves regret depending on variance of the best arm

Provides regret bounds based on first-order path-lengths

Ensures small regret in i.i.d. and other settings

Abstract

We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the variance of only the best arm; 2) a regret bound depending on the first-order path-length of only the best arm; 3) a regret bound depending on the sum of first-order path-lengths of all arms as well as an important negative term, which together lead to faster convergence rates for some normal form games with partial feedback; 4) a regret bound that simultaneously implies small regret when the best arm has small loss and logarithmic regret when there exists an arm whose expected loss is always smaller than those of others by a fixed gap (e.g. the classic i.i.d. setting). In some cases, such as the last two results, our algorithm is completely parameter-free. The main idea of our algorithm is to apply the optimism and adaptivity techniques to the well-known Online Mirror Descent framework with a special log-barrier regularizer. The challenges are to come up with appropriate optimistic predictions and correction terms in this framework. Some of our results also crucially rely on using a sophisticated increasing learning rate schedule.