Semiparametric Generalized Linear Bandits

We study finite-armed semiparametric generalized linear bandits (SGLBs), where each arm's mean reward follows a generalized linear model with an unknown common shift that may vary arbitrarily over time. This model extends semiparametric linear bandits to nonlinear links, where standard feature centering no longer removes the common shift exactly. We propose Localized Elimination for Semiparametric GLBs (**LE-SGLB**), which progressively localizes the active arms and uses centered experimental design to eliminate suboptimal arms. **LE-SGLB** achieves a leading round-dependent regret of $\tilde{\mathcal{O}}(\sqrt{dT\min \{d,\log K\}})$, up to initialization and lower-order terms, where $d$ is the feature dimension, $K$ is the number of arms, and $T$ is the horizon. Importantly, this leading term is free of the global inverse-curvature parameter $\kappa$ and is independent of the magnitude and variation of the common shifts. For logistic SGLBs, we further prove an $\Omega (d\sqrt{T})$ lower bound for $K=2^d$, equivalently $\Omega (\sqrt{dT\log K})$ in this regime, matching the leading term of our upper bound up to logarithmic factors. Moreover, the lower bound remains $\Omega (d\sqrt{T})$ even as ${\kappa }^{\star }$ grows to order $\sqrt{T}/d$, where ${\kappa }^{\star }$ denotes the inverse average slope of the logistic link at the optimal arm. Thus, the favorable $1/\sqrt{{\kappa }^{\star }}$ local-curvature dependence of ordinary logistic bandits does not extend to SGLBs.