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# Properly Learning Poisson Binomial Distributions in Almost Polynomial Time

*29th Annual Conference on Learning Theory*, PMLR 49:850-878, 2016.

#### Abstract

We give an algorithm for properly learning Poisson binomial distributions. A Poisson binomial distribution (PBD) of order n ∈\mathbbZ_+ is the discrete probability distribution of the sum of n mutually independent Bernoulli random variables. Given \widetildeO(1/ε^2) samples from an unknown PBD P, our algorithm runs in time (1/\eps)^O(\log \log (1/ε)), and outputs a hypothesis PBD that is ε-close to P in total variation distance. The sample complexity of our algorithm is known to be nearly-optimal, up to logarithmic factors, as established in previous work. However, the previously best known running time for properly learning PBDs was (1/ε)^O(\log(1/ε)), and was essentially obtained by enumeration over an appropriate ε-cover. We remark that the running time of this cover-based approach cannot be improved, as any ε-cover for the space of PBDs has size (1/ε)^Ω(\log(1/ε)). As one of our main contributions, we provide a novel structural characterization of PBDs, showing that any PBD P is ε-close to another PBD Q with O(\log(1/ε)) distinct parameters. More precisely, we prove that, for all ε>0, there exists an explicit collection \calM of (1/ε)^O(\log \log (1/ε)) vectors of multiplicities, such that for any PBD P there exists a PBD Q with O(\log(1/ε)) distinct parameters whose multiplicities are given by some element of \cal M, such that Q is ε-close to P. Our proof combines tools from Fourier analysis and algebraic geometry. Our approach to the proper learning problem is as follows: Starting with an accurate non-proper hypothesis, we fit a PBD to this hypothesis. This fitting problem can be formulated as a natural polynomial optimization problem. Our aforementioned structural characterization allows us to reduce the corresponding fitting problem to a collection of (1/ε)^O(\log \log(1/ε)) systems of low-degree polynomial inequalities. We show that each such system can be solved in time (1/ε)^O(\log \log(1/ε)), which yields the overall running time of our algorithm.