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We consider unregularized robust M-estimators for linear models underGaussian design and heavy-tailed noise, in the proportional asymptotics regimewhere the sample size n and the number of features p are both increasing suchthat $p/n \to \gamma\in (0,1)$. An estimator of the out-of-sample error of arobust M-estimator is analysed and proved to be consistent for a large familyof loss functions that includes the Huber loss. As an application of thisresult, we propose an adaptive tuning procedure of the scale parameter$\lambda>0$ of a given loss function $\rho$: choosing$\hat \lambda$ in a giveninterval $I$ that minimizes the out-of-sample error estimate of the M-estimatorconstructed with loss $\rho_\lambda(\cdot) = \lambda^2 \rho(\cdot/\lambda)$leads to the optimal out-of-sample error over $I$. The proof relies on asmoothing argument: the unregularized M-estimation objective function isperturbed, or smoothed, with a Ridge penalty that vanishes as $n\to+\infty$,and show that the unregularized M-estimator of interest inherits properties ofits smoothed version.

Error estimation and adaptive tuning for unregularized robust M-estimator