On Suboptimality of Least Squares with Application to Estimation of Convex Bodies

Gil Kur; Alexander Rakhlin; Adityanand Guntuboyina

On Suboptimality of Least Squares with Application to Estimation of Convex Bodies

Statistics Theory 2020-06-09 v1 Computational Geometry Machine Learning Metric Geometry Statistics Theory

Authors: Gil Kur , Alexander Rakhlin , Adityanand Guntuboyina

Abstract

We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension $d\geq 6$ . Specifically, we establish that Least Squares is mimimax sub-optimal, and achieves a rate of $\tilde{\Theta}_d(n^{-2/(d-1)})$ whereas the minimax rate is $\Theta_d(n^{-4/(d+3)})$ .

Keywords

least squares empirical risk minimization sample complexity convex body estimation lower bounds minimax rate

Cite

@article{arxiv.2006.04046,
  title  = {On Suboptimality of Least Squares with Application to Estimation of Convex Bodies},
  author = {Gil Kur and Alexander Rakhlin and Adityanand Guntuboyina},
  journal= {arXiv preprint arXiv:2006.04046},
  year   = {2020}
}

Comments

To appaer in Conference on Learning Theory (COLT) 2020

On Suboptimality of Least Squares with Application to Estimation of Convex Bodies

Abstract

Keywords

Cite

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