On the minimax rate of the Gaussian sequence model under bounded convex constraints

TL;DR

This paper precisely characterizes the minimax estimation rate in Gaussian sequence models constrained by bounded convex sets, linking it to the local geometric complexity of the set, and extends results to unbounded cases.

Contribution

It provides an exact minimax rate formula based on local entropy, applicable to various convex sets, and extends the analysis to unbounded constraint sets.

Findings

Minimax risk is characterized by local entropy of the constraint set.

Explicit rates are derived for hyperrectangles, ellipses, and quadratically convex sets.

Extension of minimax rates to unbounded constraint sets with known variance.

Abstract

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set $K$. Our main result shows that the minimax risk (up to constant factors) under the squared $\ell_2$ loss is given by $\epsilon^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} \epsilon^* = \sup \bigg\{\epsilon : \frac{\epsilon^2}{\sigma^2} \leq \log M^{\operatorname{loc}}(\epsilon)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(\epsilon)$ denotes the local entropy of the set $K$, and $\sigma^2$ is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets $K$ such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known $\sigma^2$ to show that the minimax rate in that case is $\epsilon^{*2}$.