# Optimal Kullback-Leibler Aggregation in Mixture Density Estimation by Maximum Likelihood

## TL;DR

This paper analyzes the maximum likelihood estimator for mixture density estimation, establishing risk bounds and optimal rates under Kullback-Leibler loss, especially in high-dimensional and sparse settings.

## Contribution

It provides sharp oracle inequalities and optimal convergence rates for the MLE in mixture models, including sparse and high-dimensional cases.

## Key findings

- MLE attains the optimal rate ((log K)/n)^{1/2} in convex aggregation.
- Under compatibility conditions, the estimator achieves the optimal sparse rate (D log K)/n.
- Introduces nearly-D-sparse aggregation and matching lower bounds.

## Abstract

We study the maximum likelihood estimator of density of $n$ independent observations, under the assumption that it is well approximated by a mixture with a large number of components. The main focus is on statistical properties with respect to the Kullback-Leibler loss. We establish risk bounds taking the form of sharp oracle inequalities both in deviation and in expectation. A simple consequence of these bounds is that the maximum likelihood estimator attains the optimal rate $((\log K)/n)^{1/2}$, up to a possible logarithmic correction, in the problem of convex aggregation when the number $K$ of components is larger than $n^{1/2}$. More importantly, under the additional assumption that the Gram matrix of the components satisfies the compatibility condition, the obtained oracle inequalities yield the optimal rate in the sparsity scenario. That is, if the weight vector is (nearly) $D$-sparse, we get the rate $(D\log K)/n$. As a natural complement to our oracle inequalities, we introduce the notion of nearly-$D$-sparse aggregation and establish matching lower bounds for this type of aggregation.