Efficient Density Estimation via Piecewise Polynomial Approximation
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We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let p be an arbitrary distribution over an interval I which is τ-close (in total variation distance) to an unknown probability distribution q that is defined by an unknown partition of I into t intervals and t unknown degree-d polynomials specifying q over each of the intervals. We give an algorithm that draws Õ(t(d+1)/^2) samples from p, runs in time (t,d,1/), and with high probability outputs a piecewise polynomial hypothesis distribution h that is (O(τ)+)-close (in total variation distance) to p. This sample complexity is essentially optimal; we show that even for τ=0, any algorithm that learns an unknown t-piecewise degree-d probability distribution over I to accuracy must use Ω(t(d+1)/(1 + (d+1))·1/^2) samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming. We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of t-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of k-monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters.
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