Learning from MOM’s principles: Le Cam’s approach
By Lecu'e Guillaume, Lerasle Matthieu. Stochastic Processes and their Applications, 2017
55 citations recorded by the source corpus.
We obtain estimation error rates for estimators obtained by aggregation of reg-ularized median-of-means tests, following a construction of Le Cam. The results hold with exponentially large probability, under only weak moments assumptions on data. 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 max minimax rate in the i.i.d. setup, number of outliers number of observations. In particular, the number of outliers may be as large as (number of data) ×(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 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.
Subfields: Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/81924f1b3394f952ce084da6ff52a31dd366436a
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Citations in the corpus
- Robust machine learning by median-of-means: Theory and practice (2017, methodology): “A modification of the ERM based on Le Cam’s estimation by tests principle [48, 49] has also been considered in [60, 50].”
- Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey (2019, unknown): “A growing body of recent work has addressed the problem of constructing regression function estimators that work well even when some of the f (X) and Y may be heavy-tailed, see Audibert and Catoni [4], Brownlees et al. [11], Catoni and Giulini [15], Chichignoud and Lederer [19], Fan et al. [28], Hsu”