On deviation probabilities in non-parametric regression with heavy-tailed noise

TL;DR

This paper investigates the limits of deviation probabilities in non-parametric regression with heavy-tailed noise, establishing fundamental bounds and proposing estimators based on median-of-means that achieve these bounds.

Contribution

It introduces a general lower bound on deviation probabilities and constructs estimators using median-of-means applied to local averaging methods, achieving optimality.

Findings

Established fundamental lower bounds on deviation probabilities.

Designed estimators that attain these bounds using median-of-means.

Demonstrated the effectiveness of median-of-means in heavy-tailed noise settings.

Abstract

This paper is devoted to the problem of determining the concentration bounds that are achievable in non-parametric regression. We consider the setting where features are supported on a bounded subset of $\mathbb{R}^d$, the regression function is Lipschitz, and the noise is only assumed to have a finite second moment. We first specify the fundamental limits of the problem by establishing a general lower bound on deviation probabilities, and then construct explicit estimators that achieve this bound. These estimators are obtained by applying the median-of-means principle to classical local averaging rules in non-parametric regression, including nearest neighbors and kernel procedures.