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We obtain estimation error rates for estimators obtained by aggregation ofregularized median-of-means tests, following a construction of Le Cam. Theresults hold with exponentially large probability -- as in the gaussianframework with independent noise- under only weak moments assumptions on dataand without assuming independence between noise and design. Any norm may beused for regularization. When it has some sparsity inducing power we recoversparse rates of convergence.  The procedure is robust since a large part of data may be corrupted, theseoutliers have nothing to do with the oracle we want to reconstruct. Our generalrisk bound is of order \begin{equation*} \max\left(\mbox{minimax rate in thei.i.d. setup}, \frac{\text{number of outliers}}{\text{number ofobservations}}\right) \enspace. \end{equation*}In particular, the number ofoutliers may be as large as (number of data) $\times$(minimax rate) withoutaffecting this rate. The other data do not have to be identically distributedbut should only have equivalent $L^1$ and $L^2$ moments.  For example, the minimax rate $s \log(ed/s)/N$ of recovery of a $s$-sparsevector in $\mathbb{R}^d$ is achieved with exponentially large probability by amedian-of-means version of the LASSO when the noise has $q_0$ moments for some$q_0>2$, the entries of the design matrix should have $C_0\log(ed)$ moments andthe dataset can be corrupted up to $C_1 s \log(ed/s)$ outliers.

Learning from MOM's principles: Le Cam's approach