Adaptive Bandit Convex Optimization with Heterogeneous Curvature

TL;DR

This paper introduces an adaptive algorithm for adversarial bandit convex optimization that dynamically adjusts to varying curvature in loss functions, outperforming previous methods especially in heterogeneous curvature scenarios.

Contribution

It develops a novel adaptive algorithm that handles unknown and heterogeneous curvature in bandit convex optimization, improving regret bounds in diverse settings.

Findings

Achieves near-optimal regret in heterogeneous curvature settings.

Reveals that some non-strongly convex functions can be effectively managed.

Extends existing frameworks to bandit feedback with innovative techniques.

Abstract

We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several homogeneous settings, but also leads to surprising results for some heterogeneous settings -- for example, while Hazan and Levy (2014) showed that $\widetilde{O}(d^{3/2}\sqrt{T})$ regret is achievable for a sequence of $T$ smooth and strongly convex $d$-dimensional functions, our algorithm reveals that the same is achievable even if $T^{3/4}$ of them are not strongly convex, and sometimes even if a constant fraction of them are not strongly convex. Our approach is inspired by the framework of Bartlett et al. (2007) who studied a similar heterogeneous setting but with stronger gradient feedback. Extending their framework to the bandit feedback setting requires novel ideas such as lifting the feasible domain and using a logarithmically homogeneous self-concordant barrier regularizer.