Efficient Minimax Optimal Estimators For Multivariate Convex Regression

05/06/2022
by   Gil Kur, et al.
0

We study the computational aspects of the task of multivariate convex regression in dimension d ≥ 5. We present the first computationally efficient minimax optimal (up to logarithmic factors) estimators for the tasks of (i) L-Lipschitz convex regression (ii) Γ-bounded convex regression under polytopal support. The proof of the correctness of these estimators uses a variety of tools from different disciplines, among them empirical process theory, stochastic geometry, and potential theory. This work is the first to show the existence of efficient minimax optimal estimators for non-Donsker classes that their corresponding Least Squares Estimators are provably minimax sub-optimal; a result of independent interest.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/03/2020

Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators

The least squares estimator (LSE) is shown to be suboptimal in squared e...
research
06/07/2020

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

We develop a technique for establishing lower bounds on the sample compl...
research
09/05/2020

Isotonic regression with unknown permutations: Statistics, computation, and adaptation

Motivated by models for multiway comparison data, we consider the proble...
research
11/09/2021

Practical, Provably-Correct Interactive Learning in the Realizable Setting: The Power of True Believers

We consider interactive learning in the realizable setting and develop a...
research
03/06/2014

Minimax Optimal Bayesian Aggregation

It is generally believed that ensemble approaches, which combine multipl...
research
10/24/2019

ERM and RERM are optimal estimators for regression problems when malicious outliers corrupt the labels

We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk ...
research
06/22/2022

Ensemble minimaxity of James-Stein estimators

This article discusses estimation of a multivariate normal mean based on...

Please sign up or login with your details

Forgot password? Click here to reset