{"id":667503,"date":"2020-06-16T11:45:15","date_gmt":"2020-06-16T18:45:15","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=667503"},"modified":"2020-06-16T11:45:15","modified_gmt":"2020-06-16T18:45:15","slug":"strong-self-concordance-and-sampling","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/strong-self-concordance-and-sampling\/","title":{"rendered":"Strong Self-Concordance and Sampling"},"content":{"rendered":"<p>Motivated by the Dikin walk, we develop aspects of an interior-point theory for sampling in high dimension. Specifically, we introduce symmetric and strong self-concordance. These properties imply that the corresponding Dikin walk mixes in \u00d5(n\u03bd\u00af) steps from a warm start in a convex body in Rn using a strongly self-concordant barrier with symmetric self-concordance parameter \u03bd\u00af. For many natural barriers, \u03bd\u00af is roughly bounded by \u03bd, the standard self-concordance parameter. We show that this property and strong self-concordance hold for the Lee-Sidford barrier. As a consequence, we obtain the first walk to mix in \u00d5(n2) steps for an arbitrary polytope in R<sup>n<\/sup>. Strong self-concordance for other barriers leads to an interesting (and unexpected) connection &#8212; for the universal and entropic barriers, it is implied by the KLS conjecture.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Motivated by the Dikin walk, we develop aspects of an interior-point theory for sampling in high dimension. Specifically, we introduce symmetric and strong self-concordance. These properties imply that the corresponding Dikin walk mixes in \u00d5(n\u03bd\u00af) steps from a warm start in a convex body in Rn using a strongly self-concordant barrier with symmetric self-concordance parameter [&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_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"STOC 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