{"id":327173,"date":"2016-11-26T19:32:42","date_gmt":"2016-11-27T03:32:42","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=327173"},"modified":"2018-10-16T21:22:40","modified_gmt":"2018-10-17T04:22:40","slug":"playing-games-approximation-algorithms","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/playing-games-approximation-algorithms\/","title":{"rendered":"Playing Games with Approximation Algorithms"},"content":{"rendered":"
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In an online linear optimization problem, on each period t, an online algorithm chooses st<\/sub> \u2208 S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt<\/sub> \u2208 R, and the algorithm incurs cost c(st<\/sub>,wt<\/sub>), where c<\/i> is a fixed cost function that is linear in the weight vector. In the full-information<\/i> setting, the vector wt<\/sub> is then revealed to the algorithm, and in the bandit<\/i> setting, only the cost experienced, c(st<\/sub>,wt<\/sub>), is revealed. The goal of the online algorithm is to perform nearly as well as the best fixed s \u2208 S in hindsight. Many repeated decision-making problems with weights fit naturally into this framework, such as online shortest-path, online TSP, online clustering, and online weighted set cover.<\/p>\n

Previously, it was shown how to convert any efficient exact<\/i> offline optimization algorithm for such a problem into an efficient online bandit algorithm in both the full-information and the bandit settings, with average cost nearly as good as that of the best fixed s \u2208 S in hindsight. However, in the case where the offline algorithm is an approximation algorithm with ratio \u03b1 > 1, the previous approach only worked for special types of approximation algorithms. We show how to convert any offline approximation algorithm for a linear optimization problem into a corresponding online approximation algorithm, with a polynomial blowup in runtime. If the offline algorithm has an \u03b1-approximation guarantee, then the expected cost of the online algorithm on any sequence is not much larger than \u03b1 times that of the best s \u2208 S, where the best is chosen with the benefit of hindsight. Our main innovation is combining Zinkevich’s algorithm for convex optimization with a geometric transformation that can be applied to any approximation algorithm. Standard techniques generalize the above result to the bandit setting, except that a “Barycentric Spanner” for the problem is also (provably) necessary as input.Our algorithm can also be viewed as a method for playing largerepeated games, where one can only compute approximate<\/i> best-responses, rather than best-responses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

In an online linear optimization problem, on each period t, an online algorithm chooses st \u2208 S from a fixed (possibly infinite) set S of feasible decisions. Nature (who may be adversarial) chooses a weight vector wt \u2208 R, and the algorithm incurs cost c(st,wt), where c is a fixed cost function that is linear […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"ACM Press","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"STOC '07 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing","msr_editors":"","msr_how_published":"","msr_isbn":"978-1-59593-631-8","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"546-555","msr_page_range_start":"546","msr_page_range_end":"555","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC '07 Proceedings of the thirty-ninth annual ACM symposium on Theory of 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