{"id":167469,"date":"2014-09-01T00:00:00","date_gmt":"2014-09-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/stochastic-primal-dual-coordinate-method-for-regularized-empirical-risk-minimization\/"},"modified":"2018-10-16T21:23:23","modified_gmt":"2018-10-17T04:23:23","slug":"stochastic-primal-dual-coordinate-method-for-regularized-empirical-risk-minimization","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/stochastic-primal-dual-coordinate-method-for-regularized-empirical-risk-minimization\/","title":{"rendered":"Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an extension with weighted sampling probabilities on the dual variables, which has a better complexity than uniform sampling on unnormalized data. Both theoretically and empirically, we show that the SPDC method has comparable or better performance than several state-of-the-art optimization methods.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An [&hellip;]<\/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":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2014-123","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":"Yuchen Zhang","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":"2014-09-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":2014,"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":[13561,13556],"msr-publication-type":[193718],"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-167469","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-artificial-intelligence","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2014-09-01","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-TR-2014-123","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":"204685","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"spdc_paper.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/spdc_paper.pdf","id":204685,"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":204685,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/spdc_paper.pdf"}],"msr-author-ordering":[{"type":"text","value":"Yuchen Zhang","user_id":0,"rest_url":false},{"type":"user_nicename","value":"lixiao","user_id":32713,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=lixiao"}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[392777],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"techreport","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. Examples abound, such as training neural networks with stochastic gradient descent, segmenting images with submodular optimization, or efficiently searching a game tree with bandit algorithms. We aim to advance the mathematical foundations of both discrete and continuous optimization and to leverage these advances to develop new algorithms with a broad set of AI applications. 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