# Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators

arXiv: https://arxiv.org/abs/2006.02044 · HTML: https://mlsanity.com/paper/2006.02044/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 7 of 7)

- [Learning linear dynamical systems under convex constraints](https://mlsanity.com/paper/2303.15121/) (2023)
- [On the Variance, Admissibility, and Stability of Empirical Risk Minimization](https://mlsanity.com/paper/2305.18508/) (2023)
- [Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach](https://mlsanity.com/paper/2401.08978/) (2024)
- [Nonparametric inference under shape constraints: past, present and future](https://mlsanity.com/paper/2509.26040/) (2025)
- [Spectrahedral Regression](https://mlsanity.com/paper/2110.14779/) (2021)
- [Near-optimal Delta-convex Estimation of Lipschitz Functions](https://mlsanity.com/paper/2511.15615/) (2025)
- [Risk reversal for least squares estimators under nested convex constraints](https://mlsanity.com/paper/2601.16041/) (2026)

## Abstract (arXiv)

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), (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 n-2/d (up to logarithmic factors) while the minimax risk is n-4/(d+4), when d ≥ 5. 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 d ≥ 1. Some new metric entropy results for convex functions are also proved which are of independent interest.