Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression

In this paper, we study two fundamentals problems: estimation of a $d$-dimensional log-concave distribution and bounded multivariate convex regression with random design. First, we show that for all $d \ge 4$ the maximum likelihood estimators of both problems achieve an optimal risk (up to a logarithmic factor) of $\Theta_d(n^{-2/(d+1)})$ in terms of squared Hellinger distance and $L_2$ squared distance, respectively. Previously, the optimality of both these estimators was known only for $d\le 3$. We also prove that the $\epsilon$-entropy numbers of the two aforementioned families are equal up to logarithmic factors. We complement these results by proving a sharp bound $\Theta_d(n^{-2/(d+4)})$ on the minimax rate (up to logarithmic factors) with respect to the total variation distance. Finally, we prove that estimating a log-concave density---even a uniform distribution on a convex set---up to a fixed accuracy requires \emph{at least} a number of samples which is exponential in the dimension. We do that by improving the dimensional constant in the best known lower bound for the minimax rate from $2^{-d}\cdot n^{-2/(d+1)}$ to $c\cdot n^{-2/(d+1)}$.