An Efficient Minimax Optimal Estimator For Multivariate Convex Regression

TL;DR

This paper introduces the first computationally efficient estimators for multivariate convex regression in higher dimensions that are nearly minimax optimal, addressing a key challenge in statistical estimation.

Contribution

It presents the first polynomial-time, minimax-optimal estimators for multivariate convex regression under Lipschitz and boundedness constraints in dimensions five and above.

Findings

Establishes minimax optimality of the proposed estimators.

Provides polynomial runtime algorithms for high-dimensional convex regression.

Addresses non-Donsker classes where least-squares estimators are suboptimal.

Abstract

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.