Efficient Best-of-Both-Worlds Algorithms for Contextual Combinatorial Semi-Bandits

TL;DR

This paper presents a novel algorithm for contextual combinatorial semi-bandits that achieves optimal regret in both adversarial and stochastic regimes, while also significantly improving computational efficiency for large-scale applications.

Contribution

It introduces a best-of-both-worlds algorithm with efficient implementation by transforming high-dimensional projections into single-variable root-finding problems.

Findings

Achieves $ ilde{O}(\sqrt{T})$ regret in adversarial regime.

Achieves $ ilde{O}(\ln T)$ regret in stochastic regime.

Provides substantial per-round speed-ups in empirical tests.

Abstract

We introduce the first best-of-both-worlds algorithm for contextual combinatorial semi-bandits that simultaneously guarantees $\widetilde{\mathcal{O}}(\sqrt{T})$ regret in the adversarial regime and $\widetilde{\mathcal{O}}(\ln T)$ regret in the corrupted stochastic regime. Our approach builds on the Follow-the-Regularized-Leader (FTRL) framework equipped with a Shannon entropy regularizer, yielding a flexible method that admits efficient implementations. Beyond regret bounds, we tackle the practical bottleneck in FTRL (or, equivalently, Online Stochastic Mirror Descent) arising from the high-dimensional projection step encountered in each round of interaction. By leveraging the Karush-Kuhn-Tucker conditions, we transform the $K$-dimensional convex projection problem into a single-variable root-finding problem, dramatically accelerating each round. Empirical evaluations demonstrate that this combined strategy not only attains the attractive regret bounds of best-of-both-worlds algorithms but also delivers substantial per-round speed-ups, making it well-suited for large-scale, real-time applications.