Efficient Minimax Optimal Estimators For Multivariate Convex Regression

We study the computational aspects of the task of multivariate convex regression in dimension $d \geq 5$. We present the first computationally efficient minimax optimal (up to logarithmic factors) estimators for the tasks of (i) $L$-Lipschitz convex regression (ii) $\Gamma$-bounded convex regression under polytopal support. The proof of the correctness of these estimators uses a variety of tools from different disciplines, among them empirical process theory, stochastic geometry, and potential theory. This work is the first to show the existence of efficient minimax optimal estimators for non-Donsker classes that their corresponding Least Squares Estimators are provably minimax sub-optimal; a result of independent interest.