{"id":145222,"date":"2005-01-01T00:00:00","date_gmt":"2005-01-01T00:00:00","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/msr-research-item\/every-decision-tree-has-an-influential-variable\/"},"modified":"2018-10-16T20:12:44","modified_gmt":"2018-10-17T03:12:44","slug":"every-decision-tree-has-an-influential-variable","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/every-decision-tree-has-an-influential-variable\/","title":{"rendered":"Every decision tree has an influential variable"},"content":{"rendered":"

We prove that for any decision tree calculating a boolean function f<\/span>:<\/span>{<\/span>\u2212<\/span>1<\/span>,<\/span>1<\/span>}<\/span>n<\/span><\/span>\u2192<\/span>{<\/span>\u2212<\/span>1<\/span>,<\/span>1<\/span>}<\/span><\/span><\/span><\/span>,<\/p>\n

\\Var[f]\u00a0\\le\u00a0\\sum_{i=1}^n\u00a0\\delta_i\u00a0\\Inf_i(f),<\/span><\/span><\/span><\/div>\n

where \u03b4<\/span>i<\/span><\/span><\/span><\/span><\/span> is the probability that the i<\/span><\/span><\/span><\/span>th input variable is read and $\\Inf_i(f)$<\/span><\/span><\/span> is the influence of the i<\/span><\/span><\/span><\/span>th variable on f<\/span><\/span><\/span><\/span>. The variance, influence and probability are taken with respect to an arbitrary product measure on {<\/span>\u2212<\/span>1<\/span>,<\/span>1<\/span>}<\/span>n<\/span><\/span><\/span><\/span><\/span>. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth d<\/span><\/span><\/span><\/span> has a variable with influence at least 1<\/span>d<\/span><\/span><\/span><\/span><\/span>. The only previous nontrivial lower bound known was \u03a9<\/span>(<\/span>d<\/span>2<\/span>\u2212<\/span>d<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>. Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least \u03a9<\/span>(<\/span>v<\/span>4<\/span>\/<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\/<\/span><\/span><\/span>p<\/span>1<\/span>\/<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>, where v<\/span><\/span><\/span><\/span> is the number of vertices and $p \\leq \\half$<\/span><\/span><\/span> is the critical threshold probability. This supersedes the milestone \u03a9<\/span>(<\/span>v<\/span>4<\/span>\/<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> bound of Hajnal and is sometimes superior to the best known lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.<\/p>\n","protected":false},"excerpt":{"rendered":"

We prove that for any decision tree calculating a boolean function f:{\u22121,1}n\u2192{\u22121,1}, \\Var[f]\u00a0\\le\u00a0\\sum_{i=1}^n\u00a0\\delta_i\u00a0\\Inf_i(f), where \u03b4i is the probability that the ith input variable is read and $\\Inf_i(f)$ is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {\u22121,1}n. It follows that […]<\/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":"Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS)","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":"Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS)","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Ryan O'Donnell, Mike Saks, Oded Schramm, Rocco 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