Near-Optimal Density Estimation in Near-Linear Time Using Variable-Width Histograms

Let p be an unknown and arbitrary probability distribution over [0, 1). We consider the problem of density estimation, in which a learning algorithm is given i.i.d. draws from p and must (with high probability) output a hypothesis distribution that is close to p. The main contribution of this paper is a highly efficient density estimation algorithm for learning using a variable-width histogram, i.e., a hypothesis distribution with a piecewise constant probability density function. In more detail, for any k and ", we give an algorithm that makes ̃ O(k/"2) draws from p, runs in ̃ O(k/"2) time, and outputs a hypothesis distribution h that is piecewise constant with O(k log2(1/")) pieces. With high probability the hypothesis h satisfies d