The Hedge Algorithm on a Continuum

author: Walid Krichene, Department of Electrical Engineering and Computer Sciences, UC Berkeley
published: Dec. 5, 2015,   recorded: October 2015,   views: 1667

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We consider an online optimization problem on a subset S of Rn (not necessarily convex), in which a decision maker chooses, at each iteration t, a probability distribution x(t) over S, and seeks to minimize a cumulative expected loss, where each loss is a Lipschitz function revealed at the end of iteration t. Building on previous work, we propose a generalized Hedge algorithm and show a O(tlogt−−−−−√) bound on the regret when the losses are uniformly Lipschitz and S is uniformly fat (a weaker condition than convexity). Finally, we propose a generalization to the dual averaging method on the set of Lebesgue-continuous distributions over S.

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