{"id":164793,"date":"2013-05-01T00:00:00","date_gmt":"2013-05-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/on-the-complexity-analysis-of-randomized-block-coordinate-descent-methods\/"},"modified":"2018-10-16T20:38:29","modified_gmt":"2018-10-17T03:38:29","slug":"on-the-complexity-analysis-of-randomized-block-coordinate-descent-methods","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/on-the-complexity-analysis-of-randomized-block-coordinate-descent-methods\/","title":{"rendered":"On the Complexity Analysis of Randomized Block-Coordinate Descent Methods"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov&#8217;s technique (SIOPT 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one in Richtarik and Takac (Math Programming 2012). Also, we obtain a better high-probability type of iteration complexity, which improves upon the one by Richtarik and Takac by at least the amount <i>O(n\/\u03b5)<\/i>, where <i>\u03b5<\/i> is the target solution accuracy and <i>n<\/i> is the number of problem blocks. In addition, for unconstrained smooth convex minimization, we develop a new technique called randomized estimate sequence to analyze the accelerated RBCD method proposed by Nesterov (SIOPT 2012) and establish a sharper expected-value type of convergence rate.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov&#8217;s technique (SIOPT 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general [&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-2013-53","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":"Zhaosong 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