  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.
