Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities

TL;DR

This paper establishes near-optimal bounds on the number of samples needed for the maximum likelihood estimator to learn multivariate log-concave densities in high dimensions, filling a gap in understanding its efficiency.

Contribution

It provides the first finite-sample upper bound for the MLE in dimensions four and higher, showing near-optimal sample complexity for learning log-concave densities.

Findings

Sample complexity upper bound of ilde{O}_d((1/ε)^{(d+3)/2})

Lower bound of Ω_d((1/ε)^{(d+1)/2})

MLE is nearly optimal in high-dimensional density estimation

Abstract

We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\mathbb{R}^d$, for all $d \geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions. In more detail, we prove that for any $d \geq 1$ and $\epsilon>0$, given $\tilde{O}_d((1/\epsilon)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\mathbb{R}^d$, the MLE outputs a hypothesis $h$ that with high probability is $\epsilon$-close to $f_0$, in squared Hellinger loss. A sample complexity lower bound of $\Omega_d((1/\epsilon)^{(d+1)/2})$ was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an $\tilde{O}(1/\epsilon)$ factor.