Title: Learning from MOM's principles: Le Cam's approach,
Abstract: We obtain estimation error rates for estimators obtained by aggregation of
regularized median-of-means tests, following a construction of Le Cam. The
results hold with exponentially large probability -- as in the gaussian
framework with independent noise- under only weak moments assumptions on data
and without assuming independence between noise and design. Any norm may be
used for regularization. When it has some sparsity inducing power we recover
sparse rates of convergence.
The procedure is robust since a large part of data may be corrupted, these
outliers have nothing to do with the oracle we want to reconstruct. Our general
risk bound is of order \begin{equation*} \max\left(\mbox{minimax rate in the
i.i.d. setup}, \frac{\text{number of outliers}}{\text{number of
observations}}\right) \enspace. \end{equation*}In particular, the number of
outliers may be as large as (number of data) $\times$(minimax rate) without
affecting this rate. The other data do not have to be identically distributed
but should only have equivalent $L^1$ and $L^2$ moments.
For example, the minimax rate $s \log(ed/s)/N$ of recovery of a $s$-sparse
vector in $\mathbb{R}^d$ is achieved with exponentially large probability by a
median-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 and
the dataset can be corrupted up to $C_1 s \log(ed/s)$ outliers.