{"id":327518,"date":"2016-11-27T18:12:06","date_gmt":"2016-11-28T02:12:06","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=327518"},"modified":"2018-10-16T21:28:53","modified_gmt":"2018-10-17T04:28:53","slug":"learning-monotonic-linear-functions","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/learning-monotonic-linear-functions\/","title":{"rendered":"Learning Monotonic Linear Functions"},"content":{"rendered":"

Learning probabilities<\/em> (p-concepts [13]) and other real-valued concepts (regression) is an important role of machine learning. For example, a doctor may need to predict the probability of getting a disease P<\/em>[y<\/em>|x<\/em>], which depends on a number of risk factors.<\/p>\n

Generalized additive models [9] are a well-studied nonparametric model in the statistics literature, usually with monotonic link functions. However, no known efficient algorithms exist for learning such a general class. We show that regression graphs efficiently<\/em> learn such real-valued concepts, while regression trees inefficiently<\/em> learn them. One corollary is that any function E<\/em>[y<\/em>|x<\/em>]=u<\/em>(w<\/em> \u00b7 x<\/em>) for umonotonic<\/em> can be learned to arbitrarily small squared error \u03b5<\/em> in time polynomial in 1\/\u03b5<\/em>, |w<\/em>|1<\/sub>, and the Lipschitz constant of u<\/em> (analogous to a margin). The model includes, as special cases, linear and logistic regression, as well as learning a noisy half-space with a margin [5, 4].<\/p>\n

Kearns, Mansour, and McAllester [12, 15], analyzed decision trees and decision graphs as boosting algorithms for classification accuracy. We extend their analysis and the boosting analogy to the case of real-valued predictors, where a small positive correlation coefficient<\/em> can be boosted to arbitrary accuracy. Viewed as a noisy boosting algorithm [3, 10], the algorithm learns both the target function and the asymmetric noise.<\/p>\n","protected":false},"excerpt":{"rendered":"

Learning probabilities (p-concepts [13]) and other real-valued concepts (regression) is an important role of machine learning. For example, a doctor may need to predict the probability of getting a disease P[y|x], which depends on a number of risk factors. Generalized additive models [9] are a well-studied nonparametric model in the statistics literature, usually with monotonic […]<\/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":"user_nicename","value":"adum","user_id":"30834"}],"msr_publishername":"Springer Berlin Heidelberg","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"17th Annual Conference on Learning Theory, COLT 2004, Banff, Canada, July 1-4, 2004","msr_editors":"","msr_how_published":"","msr_isbn":"978-3-540-22282-8 (Print) 978-3-540-27819-1 (Online)","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"487-501","msr_page_range_start":"487","msr_page_range_end":"501","msr_series":"","msr_volume":"3120","msr_copyright":"","msr_conference_name":"17th Annual Conference on Learning Theory, COLT 2004, Banff, Canada, July 1-4, 2004","msr_doi":"10.1007\/978-3-540-27819-1_34","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2004-07-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13561,13556,13546],"msr-publication-type":[193716],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-327518","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-artificial-intelligence","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Springer Berlin Heidelberg","msr_edition":"17th Annual Conference on Learning Theory, COLT 2004, Banff, Canada, July 1-4, 2004","msr_affiliation":"","msr_published_date":"2004-07-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"487-501","msr_chapter":"","msr_isbn":"978-3-540-22282-8 (Print) 978-3-540-27819-1 (Online)","msr_journal":"","msr_volume":"3120","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":"327521","msr_publicationurl":"","msr_doi":"10.1007\/978-3-540-27819-1_34","msr_publication_uploader":[{"type":"file","title":"2004-learning_monotonic_linear_functions","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions.pdf","id":327521,"label_id":0},{"type":"file","title":"2004-learning_monotonic_linear_functions","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions.ppt","id":327527,"label_id":0},{"type":"file","title":"2004-learning_monotonic_linear_functions-long_version","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions-long_version.pdf","id":327524,"label_id":0},{"type":"doi","title":"10.1007\/978-3-540-27819-1_34","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":327527,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions.ppt"},{"id":327524,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions-long_version.pdf"},{"id":327521,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/11\/2004-Learning_Monotonic_Linear_Functions.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"adum","user_id":30834,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=adum"}],"msr_impact_theme":[],"msr_research_lab":[199563],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","related_content":[],"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/327518","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":2,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/327518\/revisions"}],"predecessor-version":[{"id":536268,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/327518\/revisions\/536268"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=327518"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=327518"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=327518"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=327518"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=327518"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=327518"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=327518"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=327518"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=327518"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=327518"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=327518"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=327518"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=327518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}