Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators

by   Gil Kur, et al.

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 ≥ 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 ≥ 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 ≥ 1. Some new metric entropy results for convex functions are also proved which are of independent interest.


page 1

page 2

page 3

page 4


Efficient Minimax Optimal Estimators For Multivariate Convex Regression

We study the computational aspects of the task of multivariate convex re...

Distribution-free properties of isotonic regression

It is well known that the isotonic least squares estimator is characteri...

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

This paper deals with nonparametric projection estimators of the drift f...

Minimax Analysis for Inverse Risk in Nonparametric Planer Invertible Regression

We study a minimax risk of estimating inverse functions on a plane, whil...

On the minimax rate of the Gaussian sequence model under bounded convex constraints

We determine the exact minimax rate of a Gaussian sequence model under b...

Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

We consider the problem of nonparametric regression when the covariate i...

Learning with Square Loss: Localization through Offset Rademacher Complexity

We consider regression with square loss and general classes of functions...