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

author: Tomer Koren, Technion - Israel Institute of Technology
published: Aug. 20, 2015,   recorded: July 2015,   views: 1763


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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.

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