Robust mean estimation under star-shaped constraints with heavy-tailed noise

TL;DR

This paper investigates robust mean estimation under star-shaped constraints with heavy-tailed noise, establishing minimax rates that depend on local entropy and contamination level, applicable to unbounded sets.

Contribution

It provides the first minimax rate analysis for robust mean estimation under star-shaped constraints with heavy tails, extending known Gaussian results.

Findings

Minimax rate depends on local entropy and contamination level.

Rate matches Gaussian case for known or sign-symmetric distributions.

Results apply to both bounded and unbounded star-shaped sets.

Abstract

We study the problem of robust mean estimation with adversarially contaminated data under star-shaped constraints in a heavy-tailed noise setting, where only a finite second moment $ \sigma ^2 $ is assumed. For a contamination level $ \varepsilon$ below some constant, we show that the minimax rate of the squared $ \ell_2 $ loss is $ \max( \delta ^{*2}, \varepsilon \sigma ^2) \wedge d^2 $ for a star-shaped set with diameter $ d $ (set $d = \infty$ if the set is unbounded), with $ \delta ^* $ determined via the local entropy $ \log M^\mathrm{ loc }(\delta ,c) $ as \begin{align*} \delta ^*:= \sup\bigg\{\delta \geq 0: N\frac{\delta ^2}{\sigma ^2}\leq \log M^\mathrm{ loc }(\delta ,c) \bigg\}, \end{align*} where $ c $ is a sufficiently large constant. Crucially, we require that the sample size satisfies $N \gtrsim \mathop{ \sup }\limits_{\delta \geq 0} \log M^\mathrm{ loc }(\delta ,c)$. We also show that the minimax rate is $ \max(\delta^{*2},\varepsilon ^2\sigma ^2) \wedge d^2 $ for known or sign-symmetric distributions, matching the rate achieved in the Gaussian case.