Sample-optimal density estimation in nearly-linear time

Abstract

We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well-approximated by piecewise polynomial functions. Let f be the density function of an arbitrary univariate distribution, and suppose that f is OPT-close in L1- distance to an unknown piecewise polynomial function with t interval pieces and degree d. For any > 0, our algorithm draws n = eO(t(d + 1)=2) samples from f, runs in time eO(n), and with probability at least 9=10 outputs an O(t)-piecewise degree-d hypothesis h that is (3+) OPT+ close to f: Our approximation factor almost matches the best known information{theoretic (but computationally inefficient) upper bound of 3: Our general algorithm yields (nearly) sample- optimal and nearly-linear time estimators for a wide range of structured distribution families over both continuous and discrete domains in a unified way. For most of our applications, these are the first sample-optimal and nearly-linear time estimators in the literature. As a consequence, our work resolves the sample and computational complexities of a broad class of inference tasks via a single meta-algorithm. Moreover, we demonstrate that our algorithm performs very well in experiments. Our algorithm consists of three levels: (i) At the top level, we employ an iterative greedy algorithm for finding a good partition of the real line into the pieces of a piecewise polynomial. (ii) For each piece, we show that the sub-problem of finding a good polynomial t on the current interval can be solved efficiently with a separation oracle method. (iii) We reduce the task of finding a separating hyperplane to a combinatorial problem and design a nearly-linear algorithm for this problem. Combining these three procedures gives a density estimation algorithm with the claimed guarantees.