Abstract
Under the usual nonparametric regression model with Gaussian errors, Least Squares Estimators (LSEs) over natural subclasses of convex functions are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random design) and (iii) convex functions supported on a polytope (in fixed design). For each of these classes, the risk of the LSE is proved to be of the order (up to logarithmic factors) while the minimax risk is , when . In addition, the first rate of convergence results (worst case and adaptive) for the unrestricted convex LSE are established in fixed design for polytopal domains for all . Some new metric entropy results for convex functions are also proved, which are of independent interest.
Funding Statement
The first author was funded by the Center for Minds, Brains and Machines, funded by NSF award CCF-1231216.
The second author was funded by NSF Grant OCA-1940270.
The third author was funded by NSF CAREER Grant DMS-1654589.
The fourth author was funded by NSF Grant DMS-1712822.
Acknowledgments
We are truly thankful to the Associate Editor and three anonymous referees for their comprehensive reviews of our earlier manuscript. Their insightful feedback significantly enhanced both the content and organization of the paper.
Citation
Gil Kur. Fuchang Gao. Adityanand Guntuboyina. Bodhisattva Sen. "Convex regression in multidimensions: Suboptimality of least squares estimators." Ann. Statist. 52 (6) 2791 - 2815, December 2024. https://doi.org/10.1214/24-AOS2445
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