{"id":505307,"date":"2018-09-10T18:52:22","date_gmt":"2018-09-11T01:52:22","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=505307"},"modified":"2018-10-16T20:21:25","modified_gmt":"2018-10-17T03:21:25","slug":"revisiting-perceptron-efficient-and-label-optimal-learning-of-halfspaces","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/revisiting-perceptron-efficient-and-label-optimal-learning-of-halfspaces\/","title":{"rendered":"Revisiting Perceptron: Efficient and Label-Optimal Learning of Halfspaces"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\"><span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\">It has been a long-standing problem to efficiently learn a halfspace using as few labels as possible in the presence of noise. In this work, we propose an efficient Perceptron-based algorithm for actively learning homogeneous halfspaces under the uniform distribution over the unit sphere. Under the bounded noise condition~\\cite{MN06}, where each label is flipped with probability at most <\/span>\\(\\eta &lt;\\frac{1\/}{2}\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\">, our algorithm achieves a near-optimal label complexity of \\tilde{<\/span>\\({O}}_{}(\\frac{d\/}{(1-2\\eta {)^}_{2 }}ln\\frac{1\/}{\\epsilon })\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\"> in time \\tilde{\\({O}}_{}\\)<\/span>\\((\\frac{{d^}_{2 \/}}{\\epsilon  (1-2\\eta {)^}_3})\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\">. Under the adversarial noise condition~\\cite{ABL14, KLS09, KKMS08}, where at most a \\tilde{<\/span>\\({\\Omega }}_{}(\\epsilon )\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\"> fraction of labels can be flipped, our algorithm achieves a near-optimal label complexity of \\tilde{<\/span>\\({O}}_{}(dln\\frac{1\/}{\\epsilon })\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\"> in time \\tilde{<\/span>\\({O}}_{}(\\frac{{d^2\/}_{}}{\\epsilon })\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\">. Furthermore, we show that our active learning algorithm can be converted to an efficient passive learning algorithm that has near-optimal sample complexities with respect to <\/span>\\(\\epsilon\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\"> and <\/span>\\(d\\)<span style=\"float: none;background-color: transparent;color: #000000;font-family: 'Lucida Grande',helvetica,arial,verdana,sans-serif;font-size: 14.4px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;line-height: 20px;text-align: left;text-decoration: none;text-indent: 0px\">.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>It has been a long-standing problem to efficiently learn a halfspace using as few labels as possible in the presence of noise. In this work, we propose an efficient Perceptron-based algorithm for actively learning homogeneous halfspaces under the uniform distribution over the unit sphere. Under the bounded noise condition~\\cite{MN06}, where each label is flipped with [&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":[{"type":"text","value":"Songbai Yan","user_id":0},{"type":"user_nicename","value":"Chicheng Zhang","user_id":"36870"}],"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":"Neural Information Processing Systems 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