The minimax risk of truncated series estimators for symmetric convex polytopes

TL;DR

This paper investigates the minimax risk of truncated series estimators for symmetric convex polytopes, establishing bounds that relate the estimator's performance to the geometric complexity of the polytope.

Contribution

It introduces the approximation radius and connects it to Kolmogorov width, providing the first bounds for general convex bodies and advancing understanding of estimator optimality.

Findings

The optimal truncated series estimator is within an $O(\log m)$ factor of the best possible for polytopes defined by $m$ hyperplanes.

The approximation radius offers a new way to lower bound minimax risk based on volume.

A novel duality relationship between Kolmogorov width and the approximation radius is established.

Abstract

We study the optimality of the minimax risk of truncated series estimators for symmetric convex polytopes. We show that the optimal truncated series estimator is within $O(\log m)$ factor of the optimal if the polytope is defined by $m$ hyperplanes. This represents the first such bounds towards general convex bodies. In proving our result, we first define a geometric quantity, called the \emph{approximation radius}, for lower bounding the minimax risk. We then derive our bounds by establishing a connection between the approximation radius and the Kolmogorov width, the quantity that provides upper bounds for the truncated series estimator. Besides, our proof contains several ingredients which might be of independent interest: 1. The notion of approximation radius depends on the volume of the body. It is an intuitive notion and is flexible to yield strong minimax lower bounds; 2. The connection between the approximation radius and the Kolmogorov width is a consequence of a novel duality relationship on the Kolmogorov width, developed by utilizing some deep results from convex geometry.