Out-of-sample error estimate for robust M-estimators with convex penalty

• URL: http://arxiv.org/abs/2008.11840v5
• Date: Thu, 30 Mar 2023 17:41:03 GMT
• Title: Out-of-sample error estimate for robust M-estimators with convex penalty
• Authors: Pierre C Bellec
• Abstract summary: A generic out-of-sample error estimate is proposed for robust $M$-estimators regularized with a convex penalty. General differentiable loss functions $psi$ are allowed provided that $psi=rho'$ is 1-Lipschitz.
• Score: 5.33024001730262
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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