Improved Optimistic Algorithms for Logistic Bandits

The generalized linear bandit framework has at tracted a lot of attention in recent years by extending the well-understood linear setting and allowing to model richer reward structures. It notably covers the logistic model, widely used when rewards are binary. For logistic bandits, the frequentistregret guarantees of existing algorithms are (Formula present), where is a problem dependent constant. Unfortunately, can be arbitrarily large as it scales exponentially with the size of the decision set. This may lead to significantly loose regret bounds and poor empirical performance. In this work, we study the logistic bandit with a focus on the prohibitive dependencies introduced by. We propose a new optimistic algorithm based on a finer examination of the non-linearities of the reward function. We show that it enjoys a (Formula present) regret with no de pendency in, but for a second order term. Our analysis is based on a new tail-inequality for self normalized martingales, of independent interest.