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

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