Minimax Euclidean Separation Rates for Testing Convex Hypotheses in $\mathbb{R}^d$

TL;DR

This paper investigates the minimal Euclidean distance needed to distinguish convex hypotheses in high-dimensional Gaussian models, analyzing how this separation rate depends on dimension and sample size.

Contribution

It provides new lower and upper bounds on the minimax separation rate for testing convex hypotheses in Gaussian models, considering various smoothness conditions.

Findings

Derived bounds for smooth convex sets

Derived bounds for non-smooth convex sets

Analyzed dependence on dimension and sample size

Abstract

We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a convex subset $\mathcal{C}$ of $\mathbb{R}^d$. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension $d$ and the sample size/variance parameter $n$ giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non- smooth choices for $\mathcal{C}$.