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