Learning Sums of Independent Random Variables with Sparse Collective Support
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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 X_i's are mutually independent (but not necessarily identically distributed) integer random variables with ∪_i supp(X_i) ⊆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) · a_k samples (independent of N) and runs in time poly(1/ϵ, a_k). We prove an essentially matching lower bound: if |A| = 4, then any algorithm must use Ω( 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.
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