Instance-Wise Minimax-Optimal Algorithms for Logistic Bandits

TL;DR

This paper introduces a new algorithm for logistic bandits that achieves near-optimal regret bounds by better accounting for non-linearity effects, significantly improving over previous guarantees.

Contribution

The paper presents a novel algorithm with refined analysis for logistic bandits, providing improved problem-dependent regret bounds and establishing minimax optimality.

Findings

Regret scales as (d ext{sqrt}(T/) in favorable cases.

The new bounds outperform previous guarantees by a significant margin.

Identifies two regimes of regret, linking non-linearity effects to exploration-exploitation trade-offs.

Abstract

Logistic Bandits have recently attracted substantial attention, by providing an uncluttered yet challenging framework for understanding the impact of non-linearity in parametrized bandits. It was shown by Faury et al. (2020) that the learning-theoretic difficulties of Logistic Bandits can be embodied by a large (sometimes prohibitively) problem-dependent constant $\kappa$, characterizing the magnitude of the reward's non-linearity. In this paper we introduce a novel algorithm for which we provide a refined analysis. This allows for a better characterization of the effect of non-linearity and yields improved problem-dependent guarantees. In most favorable cases this leads to a regret upper-bound scaling as $\tilde{\mathcal{O}}(d\sqrt{T/\kappa})$, which dramatically improves over the $\tilde{\mathcal{O}}(d\sqrt{T}+\kappa)$ state-of-the-art guarantees. We prove that this rate is minimax-optimal by deriving a $\Omega(d\sqrt{T/\kappa})$ problem-dependent lower-bound. Our analysis identifies two regimes (permanent and transitory) of the regret, which ultimately re-conciliates Faury et al. (2020) with the Bayesian approach of Dong et al. (2019). In contrast to previous works, we find that in the permanent regime non-linearity can dramatically ease the exploration-exploitation trade-off. While it also impacts the length of the transitory phase in a problem-dependent fashion, we show that this impact is mild in most reasonable configurations.