{"id":354425,"date":"2017-01-17T18:46:31","date_gmt":"2017-01-18T02:46:31","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=354425"},"modified":"2018-10-16T20:32:07","modified_gmt":"2018-10-17T03:32:07","slug":"multi-scale-exploration-convex-functions-bandit-convex-optimization","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/multi-scale-exploration-convex-functions-bandit-convex-optimization\/","title":{"rendered":"Multi-Scale Exploration of Convex Functions and Bandit Convex Optimization"},"content":{"rendered":"

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span>T<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span> , where n<\/span><\/span><\/span><\/span> is the dimension and T<\/span><\/span><\/span><\/span> the number of rounds. This bound is obtained by studying the dual Bayesian maximin regret via the information ratio analysis of Russo and Van Roy, and then using the multi-scale exploration to solve the Bayesian problem.<\/p>\n","protected":false},"excerpt":{"rendered":"

We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is O~(poly(n)T\u221a) , […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-354425","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"arXiv:1507.06580v1 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