{"id":168321,"date":"2014-12-01T00:00:00","date_gmt":"2014-12-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/provable-submodular-minimization-using-wolfes-algorithm\/"},"modified":"2018-10-16T20:15:05","modified_gmt":"2018-10-17T03:15:05","slug":"provable-submodular-minimization-using-wolfes-algorithm","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/provable-submodular-minimization-using-wolfes-algorithm\/","title":{"rendered":"Provable Submodular Minimization Using Wolfe&#8217;s Algorithm"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However, these algorithms are typically not practical. In 1976, Wolfe [21] proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige [3] showed how Wolfe\u2019s algorithm can be used for SFM. For general submodular functions, this Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance.<\/p>\n<p>Despite its good practical performance, very little is known about Wolfe\u2019s minimum norm algorithm theoretically. To our knowledge, the only result is an exponential time analysis due to Wolfe [21] himself. In this paper we give a maiden convergence analysis of Wolfe\u2019s algorithm. We prove that in <em>t<\/em> iterations, Wolfe\u2019s algorithm returns an <em>O<\/em>(1\/<em>t<\/em>)-approximate solution to the min-norm point on <em>any<\/em> polytope. We also prove a robust version of Fujishige\u2019s theorem which shows that an <em>O<\/em>(1\/<em>n<\/em><sup>2<\/sup>)-approximate solution to the min-norm point on the base polytope implies <em>exact<\/em> submodular minimization. As a corollary, we get the first pseudo-polynomial time guarantee for the Fujishige-Wolfe minimum norm algorithm for unconstrained submodular function minimization.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However, these algorithms are typically not practical. In 1976, Wolfe [21] proposed an algorithm to find the minimum Euclidean norm point in a [&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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13556],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168321","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":"MIT Press Cambridge, MA, USA","msr_edition":"NIPS'14 Proceedings of the 27th International Conference on Neural 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