🤖 AI Summary
This paper addresses the computational feasibility of multivariate convex regression in dimension $d geq 5$, aiming to construct an estimator with polynomial sample complexity and minimax optimality (up to logarithmic factors) for two key settings: $L$-Lipschitz convex functions and $Gamma$-bounded convex functions supported on polyhedra. Challenging the long-standing belief that “high-dimensional convex regression admits no computationally efficient minimax-optimal estimator,” we present, for the first time in a non-Donsker class, an explicit, computationally efficient, and statistically minimax-optimal estimator—while rigorously refuting the minimax optimality of least squares. Our method integrates empirical process theory, stochastic geometry, and potential analysis, yielding an algorithm based on convex hull approximation and localized risk control. The resulting estimator achieves convergence rate $O(n^{-2/(d+4)})$ with $mathrm{poly}(n)$ time complexity—the first high-dimensional convex regression solution attaining both polynomial-time computability and statistical minimax optimality.
📝 Abstract
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.
