An Efficient Minimax Optimal Estimator For Multivariate Convex Regression
This provides efficient, minimax-optimal procedures for non-Donsker classes, addressing a bottleneck in high-dimensional convex regression for statistical and machine learning applications.
The paper tackles the computational challenge of multivariate convex regression in high dimensions by introducing the first estimators that achieve minimax optimality (up to logarithmic factors) with polynomial runtime for both L-Lipschitz and Γ-bounded convex regression under polytopal support.
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 $Γ$-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.