  We study a version of adversarial classification where an adversary is
empowered to corrupt data inputs up to some distance $\varepsilon$, using tools
from variational analysis. In particular, we describe necessary conditions
associated with the optimal classifier subject to such an adversary. Using the
necessary conditions, we derive a geometric evolution equation which can be
used to track the change in classification boundaries as $\varepsilon$ varies.
This evolution equation may be described as an uncoupled system of differential
equations in one dimension, or as a mean curvature type equation in higher
dimension. In one dimension, and under mild assumptions on the data
distribution, we rigorously prove that one can use the initial value problem
starting from $\varepsilon=0$, which is simply the Bayes classifier, in order
to solve for the global minimizer of the adversarial problem for small values
of $\varepsilon$. In higher dimensions we provide a similar result, albeit
conditional to the existence of regular solutions of the initial value problem.
In the process of proving our main results we obtain a result of independent
interest connecting the original adversarial problem with an optimal transport
problem under no assumptions on whether classes are balanced or not. Numerical
examples illustrating these ideas are also presented.
