Robust mean estimation under star-shaped constraints with heavy-tailed noise
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arXiv
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| Format: | Preprint |
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2026
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| author | Peng, Tuorui Prasadan, Akshay Neykov, Matey |
| author_facet | Peng, Tuorui Prasadan, Akshay Neykov, Matey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Robust mean estimation under star-shaped constraints with heavy-tailed noise Peng, Tuorui Prasadan, Akshay Neykov, Matey Statistics Theory 62F35, 62F30 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. |
| title | Robust mean estimation under star-shaped constraints with heavy-tailed noise |
| topic | Statistics Theory 62F35, 62F30 |
| url | https://arxiv.org/abs/2604.05063 |