# Learning Mixtures of Product Distributions over Discrete Domains

journal contribution

posted on 01.01.1988, 00:00 by Jon Feldman, Ryan O'Donnell, Rocco A. ServedioWe consider the problem of learning mixtures of product distributions over discrete
domains in the distribution learning framework introduced by Kearns et al. [Proceedings of the
26th Annual Symposium on Theory of Computing (STOC), Montr´eal, QC, 1994, ACM, New York,
pp. 273–282]. We give a poly(n/ε)-time algorithm for learning a mixture of k arbitrary product
distributions over the n-dimensional Boolean cube {0, 1}n to accuracy ε, for any constant k. Previous
polynomial-time algorithms could achieve this only for k = 2 product distributions; our result answers
an open question stated independently in [M. Cryan, Learning and Approximation Algorithms for
Problems Motivated by Evolutionary Trees, Ph.D. thesis, University of Warwick, Warwick, UK, 1999]
and [Y. Freund and Y. Mansour, Proceedings of the 12th Annual Conference on Computational
Learning Theory, 1999, pp. 183–192]. We further give evidence that no polynomial-time algorithm
can succeed when k is superconstant, by reduction from a difficult open problem in PAC (probably
approximately correct) learning. Finally, we generalize our poly(n/ε)-time algorithm to learn any
mixture of k = O(1) product distributions over {0, 1, . . . , b − 1}n, for any b = O(1).