Improved Path-length Regret Bounds for Bandits

TL;DR

This paper investigates adaptive regret bounds based on path-length for bandit problems, proving some bounds are optimal and introducing new algorithms that improve these bounds, extending results to linear bandits through novel reductions.

Contribution

The paper introduces new algorithms with improved path-length regret bounds for bandits, and extends these results to linear bandits via innovative reduction techniques.

Findings

Proved the optimality of certain path-length bounds for adaptive adversaries.

Developed algorithms that outperform previous bounds for both adversarial and oblivious settings.

Extended path-length regret bounds to linear bandits using reduction to convex body chasing.

Abstract

We study adaptive regret bounds in terms of the variation of the losses (the so-called path-length bounds) for both multi-armed bandit and more generally linear bandit. We first show that the seemingly suboptimal path-length bound of (Wei and Luo, 2018) is in fact not improvable for adaptive adversary. Despite this negative result, we then develop two new algorithms, one that strictly improves over (Wei and Luo, 2018) with a smaller path-length measure, and the other which improves over (Wei and Luo, 2018) for oblivious adversary when the path-length is large. Our algorithms are based on the well-studied optimistic mirror descent framework, but importantly with several novel techniques, including new optimistic predictions, a slight bias towards recently selected arms, and the use of a hybrid regularizer similar to that of (Bubeck et al., 2018). Furthermore, we extend our results to linear bandit by showing a reduction to obtaining dynamic regret for a full-information problem, followed by a further reduction to convex body chasing. We propose a simple greedy chasing algorithm for squared 2-norm, leading to new dynamic regret results and as a consequence the first path-length regret for general linear bandit as well.