Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables

We study the structure and learnability of sums of independent integer random variables (SIIRVs). For $k \in \mathbb{Z}_{+}$, a $k$-SIIRV of order $n \in \mathbb{Z}_{+}$ is the probability distribution of the sum of $n$ independent random variables each supported on $\{0, 1, \dots, k-1\}$. We denote by ${\cal S}_{n,k}$ the set of all $k$-SIIRVs of order $n$. In this paper, we tightly characterize the sample and computational complexity of learning $k$-SIIRVs. More precisely, we design a computationally efficient algorithm that uses $\widetilde{O}(k/\epsilon^2)$ samples, and learns an arbitrary $k$-SIIRV within error $\epsilon,$ in total variation distance. Moreover, we show that the {\em optimal} sample complexity of this learning problem is $\Theta((k/\epsilon^2)\sqrt{\log(1/\epsilon)}).$ Our algorithm proceeds by learning the Fourier transform of the target $k$-SIIRV in its effective support. Its correctness relies on the {\em approximate sparsity} of the Fourier transform of $k$-SIIRVs -- a structural property that we establish, roughly stating that the Fourier transform of $k$-SIIRVs has small magnitude outside a small set. Along the way we prove several new structural results about $k$-SIIRVs. As one of our main structural contributions, we give an efficient algorithm to construct a sparse {\em proper} $\epsilon$-cover for ${\cal S}_{n,k},$ in total variation distance. We also obtain a novel geometric characterization of the space of $k$-SIIRVs. Our characterization allows us to prove a tight lower bound on the size of $\epsilon$-covers for ${\cal S}_{n,k}$, and is the key ingredient in our tight sample complexity lower bound. Our approach of exploiting the sparsity of the Fourier transform in distribution learning is general, and has recently found additional applications.