A Jointly Efficient and Optimal Algorithm for Heteroskedastic Generalized Linear Bandits with Adversarial Corruptions

TL;DR

This paper introduces HCW-GLB-OMD, an efficient algorithm for heteroskedastic generalized linear bandits with adversarial corruptions, achieving near-optimal regret bounds across diverse settings.

Contribution

The paper proposes a novel, computationally efficient algorithm that unifies and improves regret bounds for heteroskedastic GLBs under adversarial corruptions.

Findings

Achieves regret bounds close to the lower bound, demonstrating near-optimality.

Unifies various heteroskedastic bandit settings under a single framework.

Maintains computational efficiency with O(1) complexity per iteration.

Abstract

We consider the problem of heteroskedastic generalized linear bandits (GLBs) with adversarial corruptions, which subsumes various stochastic contextual bandit settings, including heteroskedastic linear bandits and logistic/Poisson bandits. We propose HCW-GLB-OMD, which consists of two components: an online mirror descent (OMD)-based estimator and Hessian-based confidence weights to achieve corruption robustness. This is computationally efficient in that it only requires ${O}(1)$ space and time complexity per iteration. Under the self-concordance assumption on the link function, we show a regret bound of $\tilde{{O}}\left( d \sqrt{\sum_t g(\tau_t) \dot{\mu}_{t,\star}} + d^2 g_{\max} \kappa + d \kappa C \right)$, where $\dot{\mu}_{t,\star}$ is the slope of $\mu$ around the optimal arm at time $t$, $g(\tau_t)$'s are potentially exogenously time-varying dispersions (e.g., $g(\tau_t) = \sigma_t^2$ for heteroskedastic linear bandits, $g(\tau_t) = 1$ for Bernoulli and Poisson), $g_{\max} = \max_{t \in [T]} g(\tau_t)$ is the maximum dispersion, and $C \geq 0$ is the total corruption budget of the adversary. We complement this with a lower bound of $\tilde{\Omega}(d \sqrt{\sum_t g(\tau_t) \dot{\mu}_{t,\star}} + d C)$, unifying previous problem-specific lower bounds. Thus, our algorithm achieves, up to a $\kappa$-factor in the corruption term, instance-wise minimax optimality simultaneously across various instances of heteroskedastic GLBs with adversarial corruptions.