The Fourier Transform of Poisson Multinomial Distributions and Its Algorithmic Applications

An (n, k)-Poisson Multinomial Distribution (PMD) is a random variable of the form X = Sigma(n)(i=1) X-i, where the X-i's are independent random vectors supported on the set of standard basis vectors in R-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 L-1-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 epsilon using a near-optimal sample size of (O) over tilde (k)(1/epsilon(2)), and runs in time (O) over tilde (k)(1/epsilon(2)) . log n. Previously, no algorithm with a poly(1/epsilon) 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 epsilon-Nash equilibrium in time n(O(k3)) . (k/epsilon)(O(k3 log(k/epsilon)/log log(k/epsilon))k-1). The best previous algorithm for this problem [DP08, DP14] had running time n((f(k)/epsilon)k), where f(k) = Omega(k(k2)), 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 [VV10, VV11] 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 epsilon-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.