{"id":354416,"date":"2017-01-17T18:36:54","date_gmt":"2017-01-18T02:36:54","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=354416"},"modified":"2018-10-16T20:31:37","modified_gmt":"2018-10-17T03:31:37","slug":"kernel-based-methods-bandit-convex-optimization","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/kernel-based-methods-bandit-convex-optimization\/","title":{"rendered":"Kernel-Based Methods For Bandit Convex Optimization"},"content":{"rendered":"

We consider the adversarial convex bandit problem and we build the first p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>T<\/span>)<\/span><\/span><\/span><\/span> -time algorithm with p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span>T<\/span><\/span>\u221a<\/span><\/span><\/span><\/span> -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>n<\/span>9.5<\/span><\/span><\/span><\/span>T<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span> -regret, and we show that a simple variant of this algorithm can be run in p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>log<\/span><\/span>(<\/span>T<\/span>)<\/span>)<\/span><\/span><\/span><\/span> -time per step at the cost of an additional p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span>T<\/span>o<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span> factor in the regret. These results improve upon the O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>n<\/span>11<\/span><\/span><\/span><\/span>T<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span> -regret and exp<\/span><\/span>(<\/span>p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>T<\/span>)<\/span>)<\/span><\/span><\/span><\/span> -time result of the first two authors, and the log<\/span><\/span>(<\/span>T<\/span>)<\/span>p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span>T<\/span><\/span>\u221a<\/span><\/span><\/span><\/span> -regret and log<\/span><\/span>(<\/span>T<\/span>)<\/span>p<\/span>o<\/span>l<\/span>y<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span> -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve O<\/span>~<\/span><\/span><\/span><\/span>(<\/span>n<\/span>1.5<\/span><\/span><\/span><\/span>T<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span> -regret, and moreover that this regret is unimprovable (the current best lower bound being \u03a9<\/span>(<\/span>n<\/span>T<\/span><\/span>\u221a<\/span>)<\/span><\/span><\/span><\/span> and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order n<\/span>3<\/span><\/span>\/<\/span><\/span><\/span>\u03f5<\/span>2<\/span><\/span><\/span><\/span><\/span> .<\/p>\n","protected":false},"excerpt":{"rendered":"

We consider the adversarial convex bandit problem and we build the first poly(T) -time algorithm with poly(n)T\u221a -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning […]<\/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":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"arXiv:1607.03084v1","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2016-07-11","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1607.03084","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193716],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-354416","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"arXiv:1607.03084v1","msr_affiliation":"","msr_published_date":"2016-07-11","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1607.03084","msr_doi":"","msr_publication_uploader":[{"type":"url","title":"https:\/\/arxiv.org\/abs\/1607.03084","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1607.03084"}],"msr-author-ordering":[{"type":"user_nicename","value":"sebubeck","user_id":33570,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=sebubeck"},{"type":"text","value":"Ronen Eldan","user_id":0,"rest_url":false},{"type":"user_nicename","value":"yile","user_id":36030,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=yile"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[392777],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","related_content":{"projects":[{"ID":392777,"post_title":"Foundations of Optimization","post_name":"foundations-of-optimization","post_type":"msr-project","post_date":"2017-07-06 09:30:53","post_modified":"2018-12-04 14:12:39","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/foundations-of-optimization\/","post_excerpt":"Optimization methods are the engine of machine learning algorithms. 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