The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications
This work addresses foundational problems in learning theory, game theory, and statistics by providing new structural insights into PMDs, with incremental but significant improvements in algorithmic efficiency and theoretical bounds.
The paper tackled the problem of understanding Poisson Multinomial Distributions (PMDs) by proving their Fourier transform is approximately sparse, leading to applications including the first computationally efficient learning algorithm for PMDs with near-optimal sample size Õ_k(1/ε²) and runtime Õ_k(1/ε²)·log n, an efficient polynomial-time approximation scheme for computing Nash equilibria in anonymous games with improved runtime, and a multivariate central limit theorem with error bound independent of n.
An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ 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 {\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications: {\bf Learning Theory.} We design the first computationally efficient learning algorithm for PMDs with respect to the total variation distance. Our algorithm learns an arbitrary $(n, k)$-PMD within variation distance $ε$ using a near-optimal sample size of $\widetilde{O}_k(1/ε^2),$ and runs in time $\widetilde{O}_k(1/ε^2) \cdot \log n.$ Previously, no algorithm with a $\mathrm{poly}(1/ε)$ runtime was known, even for $k=3.$ {\bf 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 $n^{O(k^3)} \cdot (k/ε)^{O(k^3\log(k/ε)/\log\log(k/ε))^{k-1}}.$ The best previous algorithm for this problem had running time $n^{(f(k)/ε)^k},$ where $f(k) = Ω(k^{k^2})$, for any $k>2.$ {\bf 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 completely removing the dependence on $n$ in the error bound.