Multi-scale exploration of convex functions and bandit convex optimization

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is \tildeO(\mathrmpoly(n) \sqrtT), where n is the dimension and T the number of rounds. This bound is obtained by studying the dual Bayesian maximin regret via the information ratio analysis of Russo and Van Roy, and then using the multi-scale exploration to construct a new algorithm for the Bayesian convex bandit problem.