Variance Breakdown of Huber (M)-estimators: $n/p \rightarrow m \in (1,\infty)$

A half century ago, Huber evaluated the minimax asymptotic variance in scalar location estimation, $\min_\psi \max_{F \in {\cal F}_\epsilon} V(\psi, F) = \frac{1}{I(F_\epsilon^*)}$, where $V(\psi,F)$ denotes the asymptotic variance of the $(M)$-estimator for location with score function $\psi$, and $I(F_\epsilon^*)$ is the minimal Fisher information $\min_{{\cal F}_\epsilon} I(F)$ over the class of $\epsilon$-Contaminated Normal distributions. We consider the linear regression model $Y = X\theta_0 + W$, $W_i\sim_{\text{i.i.d.}}F$, and iid Normal predictors $X_{i,j}$, working in the high-dimensional-limit asymptotic where the number $n$ of observations and $p$ of variables both grow large, while $n/p \rightarrow m \in (1,\infty)$; hence $m$ plays the role of `asymptotic number of observations per parameter estimated'. Let $V_m(\psi,F)$ denote the per-coordinate asymptotic variance of the $(M)$-estimator of regression in the $n/p \rightarrow m$ regime. Then $V_m \neq V$; however $V_m \rightarrow V$ as $m \rightarrow \infty$. In this paper we evaluate the minimax asymptotic variance of the Huber $(M)$-estimate. The statistician minimizes over the family $(\psi_\lambda)_{\lambda > 0}$ of all tunings of Huber $(M)$-estimates of regression, and Nature maximizes over gross-error contaminations $F \in {\cal F}_\epsilon$. Suppose that $I(F_\epsilon^*) \cdot m > 1$. Then $\min_\lambda \max_{F \in {\cal F}_\epsilon} V_m(\psi_\lambda, F) = \frac{1}{I(F_\epsilon^*) - 1/m}$. Strikingly, if $I(F_\epsilon^*) \cdot m \leq 1$, then the minimax asymptotic variance is $+\infty$. The breakdown point is where the Fisher information per parameter equals unity.