Information theoretic limits of robust sub-Gaussian mean estimation under star-shaped constraints

TL;DR

This paper establishes the fundamental limits for robust mean estimation in high-dimensional star-shaped sets under adversarial corruption, providing minimax rates that depend on local entropy and noise characteristics.

Contribution

It derives the minimax risk bounds for robust mean estimation under star-shaped constraints with adversarial noise, extending to unbounded sets and different noise assumptions.

Findings

Minimax risk characterized by local entropy and noise parameters.

Algorithm achieves optimal rates under known or symmetric sub-Gaussian noise.

Slower rates are established for unknown sub-Gaussian noise scenarios.

Abstract

We obtain the minimax rate for a mean location model with a bounded star-shaped set $K \subseteq \mathbb{R}^n$ constraint on the mean, in an adversarially corrupted data setting with Gaussian noise. We assume an unknown fraction $\epsilon \le 1/2-\kappa$ for some fixed $\kappa\in(0,1/2]$ of $N$ observations are arbitrarily corrupted. We obtain a minimax risk up to proportionality constants under the squared $\ell_2$ loss of $\max(\eta^{*2},\sigma^2\epsilon^2)\wedge d^2$ with \begin{align*} \eta^* = \sup \bigg\{\eta \ge 0 : \frac{N\eta^2}{\sigma^2} \leq \log \mathcal{M}_K^{\operatorname{loc}}(\eta,c)\bigg\}, \end{align*} where $\log \mathcal{M}_K^{\operatorname{loc}}(\eta,c)$ denotes the local entropy of the set $K$, $d$ is the diameter of $K$, $\sigma^2$ is the variance, and $c$ is some sufficiently large absolute constant. A variant of our algorithm achieves the same rate for settings with known or symmetric sub-Gaussian noise, with a smaller breakdown point, still of constant order. We further study the case of unknown sub-Gaussian noise and show that the rate is slightly slower: $\max(\eta^{*2},\sigma^2\epsilon^2\log(1/\epsilon))\wedge d^2$. We generalize our results to the case when $K$ is star-shaped but unbounded.