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 $\varepsilon$ 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.