Learning k-Modal Distributions via Testing

Abstract

$ \newcommand{\eps}{\epsilon} \newcommand{\poly}{\mathrm{poly}} \newcommand{\wh}[1]{{\widehat{#1}}} $ A $k$-modal probability distribution over the discrete domain $\{1,...,n\}$ is one whose histogram has at most $k$ “peaks” and “valleys.” Such distributions are natural generalizations of monotone ($k=0$) and unimodal ($k=1$) probability distributions, which have been intensively studied in probability theory and statistics. In this paper we consider the problem of learning (i.e., performing density estimation of) an unknown $k$-modal distribution with respect to the $L_1$ distance. The learning algorithm is given access to independent samples drawn from an unknown $k$-modal distribution $p$, and it must output a hypothesis distribution $\widehat{p}$ such that with high probability the total variation distance between $p$ and $\widehat{p}$ is at most $\eps.$ Our main goal is to obtain computationally efficient algorithms for this problem that use (close to) an information-theoretically optimal number of samples. We give an efficient algorithm for this problem that runs in time $\poly(k,\log(n),1/\eps)$. For $k \leq \tilde{O}( {\log n})$, the number of samples used by our algorithm is very close (within an $\tilde{O}(\log(1/\eps))$ factor) to being information-theoretically optimal. Prior to this work computationally efficient algorithms were known only for the cases $k=0,1$ (Birgé 1987, 1997). A novel feature of our approach is that our learning algorithm crucially uses a new algorithm for property testing of probability distributions as a key subroutine. The learning algorithm uses the property tester to efficiently decompose the $k$-modal distribution into $k$ (near-)monotone distributions, which are easier to learn. A preliminary version of this work appeared in the Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012).