Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators

The least squares estimator (LSE) is shown to be suboptimal in squared error loss in the usual nonparametric regression model with Gaussian errors for $d \geq 5$ for each of the following families of functions: (i) convex functions supported on a polytope (in fixed design), (ii) bounded convex functions supported on a polytope (in random design), and (iii) convex Lipschitz functions supported on any convex domain (in random design). For each of these families, the risk of the LSE is proved to be of the order $n^{-2/d}$ (up to logarithmic factors) while the minimax risk is $n^{-4/(d+4)}$, for $d \ge 5$. In addition, the first rate of convergence results (worst case and adaptive) for the full convex LSE are established for polytopal domains for all $d \geq 1$. Some new metric entropy results for convex functions are also proved which are of independent interest.