Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Statistical consistency and asymptotic normality for high-dimensional
  robust M-estimators   We study theoretical properties of regularized robust M-estimators,
applicable when data are drawn from a sparse high-dimensional linear model and
contaminated by heavy-tailed distributions and/or outliers in the additive
errors and covariates. We first establish a form of local statistical
consistency for the penalized regression estimators under fairly mild
conditions on the error distribution: When the derivative of the loss function
is bounded and satisfies a local restricted curvature condition, all stationary
points within a constant radius of the true regression vector converge at the
minimax rate enjoyed by the Lasso with sub-Gaussian errors. When an appropriate
nonconvex regularizer is used in place of an l_1-penalty, we show that such
stationary points are in fact unique and equal to the local oracle solution
with the correct support---hence, results on asymptotic normality in the
low-dimensional case carry over immediately to the high-dimensional setting.
This has important implications for the efficiency of regularized nonconvex
M-estimators when the errors are heavy-tailed. Our analysis of the local
curvature of the loss function also has useful consequences for optimization
when the robust regression function and/or regularizer is nonconvex and the
objective function possesses stationary points outside the local region. We
show that as long as a composite gradient descent algorithm is initialized
within a constant radius of the true regression vector, successive iterates
will converge at a linear rate to a stationary point within the local region.
Furthermore, the global optimum of a convex regularized robust regression
function may be used to obtain a suitable initialization. The result is a novel
two-step procedure that uses a convex M-estimator to achieve consistency and a
nonconvex M-estimator to increase efficiency.

26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con