A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families

TL;DR

This paper introduces the first polynomial-time algorithm for computing the multivariate log-concave maximum likelihood distribution, leveraging a novel connection to exponential families and convex optimization.

Contribution

It presents a novel polynomial-time algorithm for multivariate log-concave MLE using a new connection to locally exponential families and convex optimization techniques.

Findings

Algorithm runs in polynomial time in n, d, and 1/ε.

Returns a distribution with likelihood within ε of the maximum.

Establishes a new connection between log-concave distributions and exponential families.

Abstract

We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given $n$ points in $\mathbb{R}^d$ and an accuracy parameter $\epsilon>0$, runs in time $poly(n,d,1/\epsilon),$ and returns a log-concave distribution which, with high probability, has the property that the likelihood of the $n$ points under the returned distribution is at most an additive $\epsilon$ less than the maximum likelihood that could be achieved via any log-concave distribution. This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task. Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, "locally" possesses key properties of exponential families. This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method. Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.