An Efficient Minimax Optimal Estimator For Multivariate Convex Regression

This work studies the computational aspects of multivariate convex regression in dimensions $d \ge 5$. Our results include the \emph{first} estimators that are minimax optimal (up to logarithmic factors) with polynomial runtime in the sample size for both $L$-Lipschitz convex regression, and $\Gamma$-bounded convex regression under polytopal support. Our analysis combines techniques from empirical process theory, stochastic geometry, and potential theory, and leverages recent algorithmic advances in mean estimation for random vectors and in distribution-free linear regression. These results provide the first efficient, minimax-optimal procedures for non-Donsker classes for which their corresponding least-squares estimator is provably minimax-suboptimal.