{"id":438240,"date":"2017-11-06T16:20:59","date_gmt":"2017-11-07T00:20:59","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=438240"},"modified":"2018-10-16T22:31:41","modified_gmt":"2018-10-17T05:31:41","slug":"linear-convergence-frank-wolfe-type-algorithm-trace-norm-balls","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/linear-convergence-frank-wolfe-type-algorithm-trace-norm-balls\/","title":{"rendered":"Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls"},"content":{"rendered":"

We propose a rank-k<\/span><\/span><\/span><\/i><\/span> variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1<\/span><\/span><\/span><\/span> -SVD) in Frank-Wolfe with a top-k<\/span><\/span><\/span><\/i><\/span> singular-vector computation (k<\/span><\/span><\/span><\/i><\/span> -SVD), which can be done by repeatedly applying 1<\/span><\/span><\/span><\/span> -SVD k<\/span><\/span><\/span><\/i><\/span> times. Alternatively, our algorithm can be viewed as a rank-k<\/span><\/span><\/span><\/i><\/span> restricted version of projected gradient descent. We show that our algorithm has a linear convergence rate when the objective function is smooth and strongly convex, and the optimal solution has rank at most k<\/span><\/span><\/span><\/i><\/span> . This improves the convergence rate and the total time complexity of the Frank-Wolfe method and its variants.<\/p>\n","protected":false},"excerpt":{"rendered":"

We propose a rank-k variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1 -SVD) in Frank-Wolfe with a top-k singular-vector computation (k -SVD), which can be done by repeatedly applying 1 -SVD k times. Alternatively, our algorithm can be viewed as a […]<\/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":"Advances in Neural Information Processing Systems 30 (NIPS 2017)","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":"Advances in Neural Information Processing Systems 30 (NIPS 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