  Learning classifiers that are robust to adversarial examples has received a
great deal of recent attention. A major drawback of the standard robust
learning framework is the imposition of an artificial robustness radius $r$
that applies to all inputs, and ignores the fact that data may be highly
heterogeneous. In this paper, we address this limitation by proposing a new
framework for adaptive robustness, called neighborhood preserving robustness.
We present sufficient conditions under which general non-parametric methods
that can be represented as weight functions satisfy our notion of robustness,
and show that both nearest neighbors and kernel classifiers satisfy these
conditions in the large sample limit.
