Adaptive Risk Bounds in Unimodal Regression

TL;DR

This paper analyzes the statistical properties of the unimodal least squares estimator, demonstrating its adaptivity and developing a variational risk representation for non-convex parameter spaces, extending understanding beyond isotonic regression.

Contribution

It introduces a new variational risk representation for unimodal regression and establishes its adaptivity, addressing non-convexity challenges in the parameter space.

Findings

Unimodal least squares estimator is adaptive to the true sequence complexity.

Develops a variational risk representation applicable to non-convex parameter spaces.

Extends techniques from isotonic regression to unimodal regression.

Abstract

We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al(2015) and Bellec(2015). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a general variational representation of the risk that holds whenever the parameter space can be expressed as a finite union of convex sets, using techniques that may be of interest in other settings.