An Experimental Design Approach for Regret Minimization in Logistic Bandits

In this work we consider the problem of regret minimization for logistic bandits. The main challenge bandits is reducing dependence on a potentially large dependent constant that can at worst scale exponentially with norm unknown parameter vector. Previous works have applied self-concordance function to remove worst-case providing guarantees move reduce case lower order terms only polylogarithmic term and as well linear dimension parameter. This improves upon prior art by 1) removing all scaling 2) square root in fixed arm setting employing an experimental design procedure. Our bound fact takes tighter instance (i.e., gap) first time We also propose new warmup sampling algorithm dramatically general prove it term's dependency some instances. Finally, discuss impact bias MLE bandit d dimensions, example where d^2 (cf., bandits) may not be improved long used how bias-corrected estimators make closer d.