The fourier transform of poisson multinomial distributions and its algorithmic applications

Abstract

An (n, k)-Poisson Multinomial Distribution (PMD) is a random variable of the form X = ∑i=1n Xi, where the Xi’s are independent random vectors supported on the set of standard basis vectors in k. In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is approximately sparse, i.e., its L1-norm is small outside a small set. By building on this result, we obtain the following applications: Learning Theory. We give the first computationally efficient learning algorithm for PMDs under the total variation distance. Our algorithm learns an (n, k)-PMD within variation distance ε using a near-optimal sample size of Ok(1/ε2), and runs in time Ok(1/ε2) · logn. Previously, no algorithm with a (1/ε) runtime was known, even for k=3. Game Theory. We give the first efficient polynomial-time approximation scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized anonymous games with n players and k strategies, our algorithm computes a well-supported ε-Nash equilibrium in time nO(k3) · (k/ε)O(k3log(k/ε)/loglog(k/ε))k−1. The best previous algorithm for this problem had running time n(f(k)/ε)k, where f(k) = Ω(kk2), for any k>2. Statistics. We prove a multivariate central limit theorem (CLT) that relates an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance. Our new CLT strengthens the CLT of Valiant and Valiant by removing the dependence on n in the error bound. Along the way we prove several new structural results of independent interest about PMDs. These include: (i) a robust moment-matching lemma, roughly stating that two PMDs that approximately agree on their low-degree parameter moments are close in variation distance; (ii) near-optimal size proper ε-covers for PMDs in total variation distance (constructive upper bound and nearly-matching lower bound). In addition to Fourier analysis, we employ a number of analytic tools, including the saddlepoint method from complex analysis, that may find other applications.