  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.
