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CATEGORIES:Statistics
SUMMARY:Learning from MOM's principles - Guillaume LecuĂ©
(ENSAE)
DTSTART;TZID=Europe/London:20171117T160000
DTEND;TZID=Europe/London:20171117T170000
UID:TALK93397AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/93397
DESCRIPTION:(Joint work with Matthieu Lerasle)\nWe obtain theo
retical and practical performances for median of m
eans estimators. \n\nFrom a theoretical point of v
iew\, estimation and prediction error bounds achie
ved by the MOM estimators hold with exponentially
large probability -- as in the gaussian framework
with independent noise-- under only weak moments a
ssumptions on the data and without assuming indepe
ndence between the noise and the design. Moreover\
, MOM procedures are robust since a large part of
the data may have nothing to do with the oracle w
e want to reconstruct. Our general risk bound is o
f order max(minimax rate of convergence in the i.i
.d. setup\, (number of outliers)/number of observa
tions)). In particular\, the number of outliers m
ay be as large as (number of data)*(minimax rate)
without affecting the statistical properties of t
he MOM estimator. \n\nA regularization norm may al
so be used together with the MOM criterium. In tha
t case\, any norm can be used for regularization.
When it has some sparsity inducing power we recove
r sparse rates of convergence and sparse oracle in
equalities. For example\, the minimax rate s log(d
/s)/N of recovery of a s-sparse vector in R^d is a
chieved by a median-of-means version of the LASSO
when the noise has q_0 moments for some q_0>2\, th
e design matrix C_0\\log(d) moments and the datase
t is corrupted by s log(d/s) outliers. This result
holds with exponentially large probability as if
the noise and the design were i.i.d. Gaussian rand
om variables. \n\nOn the practical side\, MOM esti
mators (and their associated regularized versions)
can easily be implemented. Actually\, most gradie
nt descent algorithms used to implement (non-robus
t) estimators like the LASSO can easily be transfo
rmed into a robust one by using a MOM approach.
LOCATION:MR12
CONTACT:Quentin Berthet
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