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

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 $\epsilon$ below some constant, we show that the minimax rate of the squared ${\mathrm{\ell }}_2$ loss is $\max ({\delta }^{\ast 2},\epsilon {\sigma }^2)\wedge d^2$ for a star-shaped set with diameter $d$ (set $d=\mathrm{\infty }$ if the set is unbounded), with ${\delta }^{\ast }$ determined via the local entropy $\log M^{\mathrm{l}\mathrm{o}\mathrm{c}}(\delta ,c)$ as \begin{align*} δ^:= \sup\bigg{δ\geq 0: N\frac{δ^2}{σ^2}\leq \log M^$\mathrm{l}\mathrm{o}\mathrm{c}$(δ,c) \bigg}, \end{align} where $c$ is a sufficiently large constant. Crucially, we require that the sample size satisfies $N\gtrsim \underset{\delta \ge 0}{}\log M^{\mathrm{l}\mathrm{o}\mathrm{c}}(\delta ,c)$. We also show that the minimax rate is $\max ({\delta }^{\ast 2},{\epsilon }^2{\sigma }^2)\wedge d^2$ for known or sign-symmetric distributions, matching the rate achieved in the Gaussian case.