{"id":327548,"date":"2016-11-27T18:47:42","date_gmt":"2016-11-28T02:47:42","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=327548"},"modified":"2018-10-16T20:25:05","modified_gmt":"2018-10-17T03:25:05","slug":"noise-tolerant-learning-parity-problem-statistical-query-model","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/noise-tolerant-learning-parity-problem-statistical-query-model\/","title":{"rendered":"Noise-Tolerant Learning, the Parity Problem, and the Statistical Query Model"},"content":{"rendered":"<p>We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise, a problem closely related to several cryptographic and coding problems. Our algorithm runs in polynomial time for the case of parity functions that depend on only the first <i>O<\/i>(log <i>n<\/i> log log <i>n<\/i>) bits of input, which provides the first known instance of an efficient noise-tolerant algorithm for a concept class that is not learnable in the Statistical Query model of Kearns [1998]. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model.In coding-theory terms, what we give is a poly(<i>n<\/i>)-time algorithm for decoding linear <i>k<\/i> \u00d7 <i>n<\/i> codes in the presence of random noise for the case of <i>k<\/i> = <i>c<\/i> log <i>n<\/i> log log <i>n<\/i> for some <i>c<\/i> > 0. (The case of <i>k<\/i> = <i>O<\/i>(log <i>n<\/i>) is trivial since one can just individually check each of the 2<sup><i>k<\/i><\/sup> possible messages and choose the one that yields the closest codeword.)A natural extension of the statistical query model is to allow queries about statistical properties that involve <i>t<\/i>-tuples of examples, as opposed to just single examples. The second result of this article is to show that any class of functions learnable (strongly or weakly) with <i>t<\/i>-wise queries for <i>t<\/i> = <i>O<\/i>(log <i>n<\/i>) is also weakly learnable with standard unary queries. Hence, this natural extension to the statistical query model does not increase the set of weakly learnable functions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise, a problem closely related to several cryptographic and coding problems. Our algorithm runs in polynomial time for the case of parity functions that depend on only the first O(log n log log n) bits of input, which [&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":"ACM Press","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"4","msr_journal":"Journal of the ACM 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