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

We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the emph{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 (varepsilonu003e0), given (widetilde{O}_d((1/varepsilon)^{(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 (varepsilon)-close to (f_0), in squared Hellinger loss. A sample complexity lower bound of (Omega_d((1/varepsilon)^{(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/varepsilon)) factor.