Minimax Rates for Robust Gaussian Mean Estimation Near the Breakdown Point under Constraints

@misc{peng2026minimax,
  title = {{Minimax Rates for Robust Gaussian Mean Estimation Near the Breakdown Point under Constraints}},
  author = {Tuorui Peng and Matey Neykov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36283},
  eprint = {2610.36283},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36283},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Minimax Rates for Robust Gaussian Mean Estimation Near the Breakdown Point under Constraints
AU  - Tuorui Peng
AU  - Matey Neykov
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36283
UR  - https://arxiv.org/abs/2610.36283
AB  - We study the mean estimation of Gaussian location model with star-shaped constraints under adversarial contamination, where at most $ k<N/2 $ among $ N $ observations are corrupted by an oracle adversary. We characterize the minimax squared $ \ell_2 $ risk throughout the full range of contamination level $ \varepsilon = k/N \in [0,1/2) $. The minimax risk is Masymp ( δ_ε ^*2 ∨ σ^2ε ^2log e/(1−2ε ) ) ∧ d^2 , where $ δ_\varepsilon ^* := \sup \big\{ r \geq 0: (1-2\varepsilon )\frac{Nr ^2}{σ^2} \leq \log M^\mathrm{ loc }_K(r ,c) \big\} $, $ \log M^\mathrm{ loc }_K(r ,c) $ is the local entropy of constraint set $ K $, $ d $ is the diameter of $ K $ (set $ d=\infty $ if $ K $ is unbounded), $ σ^2 I_n $ is the covariance of the Gaussian noise, and $ c $ is a sufficiently large constant. Our result sheds light on the interaction between contamination error and information theoretic complexity in the regime of proximity to the breakdown point $ \varepsilon \uparrow 1/2 $.
ER  -