{"id":392975,"date":"2015-02-23T00:00:47","date_gmt":"2015-02-23T08:00:47","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=392975"},"modified":"2018-10-16T20:12:42","modified_gmt":"2018-10-17T03:12:42","slug":"bandit-convex-optimization-sqrtt-regret-one-dimension","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/bandit-convex-optimization-sqrtt-regret-one-dimension\/","title":{"rendered":"Bandit Convex Optimization: sqrt{T} Regret in One Dimension"},"content":{"rendered":"

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 \u0398\u02dc(\u221aT)<\/em> 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 \u0398\u02dc(\u221aT) 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 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