Learning sums of independent random variables with sparse collective support

We study the learnability of sums of independent integer random variables given a bound on the size of the union of their supports. For A ⊂ ℤ ₊ , a sum of independent random variables with collective support A (called an A-sum in this paper) is a distribution S = X ₁ + • • • + X _N where the Xi's are mutually independent (but not necessarily identically distributed) integer random variables with ∪isupp(Xi) ⊂ A. We give two main algorithmic results for learning such distributions: 1) For the case |A| = 3, we give an algorithm for learning A-sums to accuracy ϵ that uses poly(1/ϵ) samples and runs in time poly(1/ϵ), independent of N and of the elements of A. 2) For an arbitrary constant k ≥ 4, if A = {a ₁ ,⋯, a _k } with 0 < a ₁ <⋯ < a _k , we give an algorithm that uses poly(1/ϵ) • log log a _k samples (independent of N) and runs in time poly(1/ϵ, loga _k ). We prove an essentially matching lower bound: if |A| = 4, then any algorithm must use Ω(log log a ₄ ) samples even for learning to constant accuracy. We also give similar-in-spirit (but quantitatively very different) algorithmic results, and essentially matching lower bounds, for the case in which A is not known to the learner. Our learning algorithms employ new limit theorems which may be of independent interest. Our lower bounds rely on equidistribution type results from number theory. Our algorithms and lower bounds together settle the question of how the sample complexity of learning sums of independent integer random variables scales with the elements in the union of their supports, both in the known-support and unknown-support settings. Finally, all our algorithms easily extend to the "semi-agnostic" learning model, in which training data is generated from a distribution that is only cϵ-close to some A-sum for a constant c > 0.