{"id":392861,"date":"2015-07-09T00:00:26","date_gmt":"2015-07-09T07:00:26","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=392861"},"modified":"2018-10-16T20:12:30","modified_gmt":"2018-10-17T03:12:30","slug":"sampling-log-concave-distribution-projected-langevin-monte-carlo","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/sampling-log-concave-distribution-projected-langevin-monte-carlo\/","title":{"rendered":"Sampling from a log-concave distribution with Projected Langevin Monte Carlo"},"content":{"rendered":"

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in O~(n7<\/sup>)<\/em> steps (where n<\/em> is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of O~(n4<\/sup>)<\/em> was proved by Lov{\\’a}sz and Vempala.<\/p>\n","protected":false},"excerpt":{"rendered":"

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main […]<\/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],"msr-publication-type":[193715],"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-392861","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2015-07-09","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":"392867","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1507.02564","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"1507.02564","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2015\/07\/1507.02564.pdf","id":392867,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/1507.02564","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1507.02564"}],"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":"text","value":"Joseph Lehec","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[392777],"publication":[],"video":[],"download":[],"msr_publication_type":"article","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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