  We consider the problem of finding optimal classifiers in an adversarial
setting where the class-1 data is generated by an attacker whose objective is
not known to the defender -- an aspect that is key to realistic applications
but has so far been overlooked in the literature. To model this situation, we
propose a Bayesian game framework where the defender chooses a classifier with
no a priori restriction on the set of possible classifiers. The key difficulty
in the proposed framework is that the set of possible classifiers is
exponential in the set of possible data, which is itself exponential in the
number of features used for classification. To counter this, we first show that
Bayesian Nash equilibria can be characterized completely via functional
threshold classifiers with a small number of parameters. We then show that this
low-dimensional characterization enables to develop a training method to
compute provably approximately optimal classifiers in a scalable manner; and to
develop a learning algorithm for the online setting with low regret (both
independent of the dimension of the set of possible data). We illustrate our
results through simulations.
