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We analyze the minimax regret of the adversarial bandit convex optimizationproblem. Focusing on the one-dimensional case, we prove that the minimax regretis $\widetilde\Theta(\sqrt{T})$ and partially resolve a decade-old openproblem. Our analysis is non-constructive, as we do not present a concretealgorithm that attains this regret rate. Instead, we use minimax duality toreduce the problem to a Bayesian setting, where the convex loss functions aredrawn from a worst-case distribution, and then we solve the Bayesian version ofthe problem with a variant of Thompson Sampling. Our analysis features a noveluse of convexity, formalized as a "local-to-global" property of convexfunctions, that may be of independent interest.

Bandit Convex Optimization: sqrt{T} Regret in One Dimension