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

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 \begin{equation*} \mathfrak{M}\asymp \big( δ_\varepsilon ^{2} \vee σ^2\varepsilon ^2\log \frac{e}{1-2\varepsilon } \big) \wedge d^2 , \end{equation} 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 $\varepsilon \uparrow 1/2$.