A generic out-of-sample error estimate is proposed for robust $M$-estimators regularized with a convex penalty in high-dimensional linear regression where $(X,y)$ is observed and $p,n$ are of the same order. If $\psi$ is the derivative of the robust data-fitting loss $\rho$, the estimate depends on the observed data only through the quantities $\hat\psi = \psi(y-X\hat\beta)$, $X^\top \hat\psi$ and the derivatives $(\partial/\partial y) \hat\psi$ and $(\partial/\partial y) X\hat\beta$ for fixed $X$. The out-of-sample error estimate enjoys a relative error of order $n^{-1/2}$ in a linear model with Gaussian covariates and independent noise, either non-asymptotically when $p/n\le \gamma$ or asymptotically in the high-dimensional asymptotic regime $p/n\to\gamma'\in(0,\infty)$. General differentiable loss functions $\rho$ are allowed provided that $\psi=\rho'$ is 1-Lipschitz. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the $\ell_1$-penalized Huber M-estimator if the number of corrupted observations and sparsity of the true $\beta$ are bounded from above by $s_*n$ for some small enough constant $s_*\in(0,1)$ independent of $n,p$. For the square loss and in the absence of corruption in the response, the results additionally yield $n^{-1/2}$-consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty, estimates that were previously known for the Lasso.

翻译：如果$\psi是可靠数据匹配损失的衍生物 $\\ rho $,则该估计数仅取决于观测到的数据 $\ hat\ psi =\ psi =\ pssi (y- Xhat\ beta) $. $Xtop\ hat\psi 和衍生物 $(部分/部分 y)\ hat\ psi y) 在高维线性线性线性回归中 $(部分/部分) 和美元是相同的。如果美元是(部分/部分) 美元和美元 美元,则该估计数仅取决于观测到的数据量 。 当美元/n\ hat\ bet\ betati $(部分/部分/部分) 和衍生物 美元(部分) 美元(部分) 和美元(部分(部分) 美元(部分) 等离子(部分) 等离子(美元) 等(美元) 等值的數(美元)的數值變化的數(美元)的數數(美元)