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

We consider heteroskedastic generalized linear bandits (GLBs) with adversarial corruptions and propose \texttt{HCW-GLB-OMD}, a simple algorithm that combines an online mirror descent (OMD)-based estimator with Hessian-based confidence weights. Although both components are well studied, establishing guarantees for the combined algorithm requires a nontrivial analysis. We establish a corruption-robust confidence sequence using technical tools not required in prior analyses, including a weighted likelihood supermartingale and a self-bounding argument. For the regret analysis, we derive a new Hessian potential lemma to handle the interaction between adversarial corruption and adaptive confidence weighting. For self-concordant GLMs, we prove a high-probability regret bound of $\widetilde{\mathcal{O}}\bigl( d \sqrt{\sum_t g(τ_t) \dotμ_{t,\star}} + d^2 g_{\max} κ+ d(1+κ)(C\vee1) \bigr)$, where $\dotμ_{t,\star}$ is the slope of $μ$ at the optimal arm at time $t$, $g(τ_t)$ is the observed dispersion, $g_{\max}=\max_t g(τ_t)$, $κ$ is the worst-case inverse slope of $μ$, and $C$ is the corruption budget. The algorithm attains this bound with one OMD update per round and $\mathcal{O}(d^2)$ storage. We complement this guarantee with matching lower bounds for the leading $\sqrt{T}$ term and, up to $κ$, the $dC$ corruption term.