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 A ⊂Z
₊
ubset A of non-negative integers, 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 X
_i
's are mutually independent (but not necessarily identically distributed) integer random variables all of whose supports are contained in 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, log a
_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 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 c*ε-close to some A-sum for a constant c>0.