  In real world, our datasets often contain outliers. Moreover, the outliers
can seriously affect the final machine learning result. Most existing
algorithms for handling outliers take high time complexities (e.g. quadratic or
cubic complexity). {\em Coreset} is a popular approach for compressing data so
as to speed up the optimization algorithms. However, the current coreset
methods cannot be easily extended to handle the case with outliers. In this
paper, we propose a new variant of coreset technique, {\em layered sampling},
to deal with two fundamental robust optimization problems: {\em
$k$-median/means clustering with outliers} and {\em linear regression with
outliers}. This new coreset method is in particular suitable to speed up the
iterative algorithms (which often improve the solution within a local range)
for those robust optimization problems. Moreover, our method is easy to be
implemented in practice. We expect that our framework of layered sampling will
be applicable to other robust optimization problems.
