Learning Poisson Binomial Distributions (Extended Abstract)
Constantinos Daskalakis, Ilias Diakonikolas, Rocco A. Servedio · 2012
We consider a basic problem in unsupervised learning: learn-ing an unknown Poisson Binomial Distribution. A Poisson Binomial Distribution (PBD) over {0, 1,..., n} is the distri-bution of a sum of n independent Bernoulli random variables which may have arbitrary, potentially non-equal, expecta-tions. These distributions were first studied by S. Poisson in 1837 [Poi37] and are a natural n-parameter generalization of the familiar Binomial Distribution. Surprisingly, prior to our work this basic learning problem was poorly understood, and known results for it were far from optimal. We essentially settle the complexity of the learning prob-lem for this basic class of distributions. As our main re-sult we give a highly efficient algorithm which learns to ǫ-accuracy using Õ(1/ǫ3) samples independent of n. The running time of the algorithm is quasilinear in the size of its input data, i.e. Õ(log(n)/ǫ3) bit-operations1 (observe that each draw from the distribution is a log(n)-bit string). This is nearly optimal since any algorithm must use Ω(1/ǫ2) samples. We also give positive and negative results for some extensions of this learning problem.