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Academic Commons Search Resultsen-usOn Decision Trees, Influences, and Learning Monotone Decision Trees
http://academiccommons.columbia.edu/catalog/ac:109797
O'Donnell, Ryan; Servedio, Rocco Anthonyhttp://hdl.handle.net/10022/AC:P:29225Fri, 22 Apr 2011 14:25:40 +0000In this note we prove that a monotone boolean function computable by a decision tree of size $s$ has average sensitivity at most $\sqrt{\log_2 s}$. As a consequence we show that monotone functions are learnable to constant accuracy under the uniform distribution in time polynomial in their decision tree size.Computer scienceras2105Computer ScienceTechnical reportsLearning mixtures of product distributions over discrete domains
http://academiccommons.columbia.edu/catalog/ac:110398
Feldman, Jon; O'Donnell, Ryan; Servedio, Rocco Anthonyhttp://hdl.handle.net/10022/AC:P:29411Thu, 21 Apr 2011 12:41:48 +0000We consider the problem of learning mixtures of product distributions over discrete domains in the distribution learning framework introduced by Kearns et al. We give a $\poly(n/\eps)$ time algorithm for learning a mixture of $k$ arbitrary product distributions over the $n$-dimensional Boolean cube $\{0,1\}^n$ to accuracy $\eps$, for any constant $k$. Previous polynomial time algorithms could only achieve this for $k = 2$ product distributions; our result answers an open question stated independently by Cryan and by Freund and Mansour. We further give evidence that no polynomial time algorithm can succeed when $k$ is superconstant, by reduction from a notorious open problem in PAC learning. Finally, we generalize our $\poly(n/\eps)$ time algorithm to learn any mixture of $k = O(1)$ product distributions over $\{0,1, \dots, b\}^n$, for any $b = O(1)$.Computer scienceras2105Industrial Engineering and Operations Research, Computer ScienceTechnical reports