  We present a transductive learning algorithm that takes as input training
examples from a distribution $P$ and arbitrary (unlabeled) test examples,
possibly chosen by an adversary. This is unlike prior work that assumes that
test examples are small perturbations of $P$. Our algorithm outputs a selective
classifier, which abstains from predicting on some examples. By considering
selective transductive learning, we give the first nontrivial guarantees for
learning classes of bounded VC dimension with arbitrary train and test
distributions---no prior guarantees were known even for simple classes of
functions such as intervals on the line. In particular, for any function in a
class $C$ of bounded VC dimension, we guarantee a low test error rate and a low
rejection rate with respect to $P$. Our algorithm is efficient given an
Empirical Risk Minimizer (ERM) for $C$. Our guarantees hold even for test
examples chosen by an unbounded white-box adversary. We also give guarantees
for generalization, agnostic, and unsupervised settings.
