  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 $\tilde{O}(\mathrm{poly}(n) \sqrt{T})$, 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 solve the Bayesian
problem.
