  A common challenge across all areas of machine learning is that training data
is not distributed like test data, due to natural shifts, "blind spots," or
adversarial examples; such test examples are referred to as out-of-distribution
(OOD) test examples. We consider a model where one may abstain from predicting,
at a fixed cost. In particular, our transductive abstention algorithm takes
labeled training examples and unlabeled test examples as input, and provides
predictions with optimal prediction loss guarantees. The loss bounds match
standard generalization bounds when test examples are i.i.d. from the training
distribution, but add an additional term that is the cost of abstaining times
the statistical distance between the train and test distribution (or the
fraction of adversarial examples). For linear regression, we give a
polynomial-time algorithm based on Celis-Dennis-Tapia optimization algorithms.
For binary classification, we show how to efficiently implement it using a
proper agnostic learner (i.e., an Empirical Risk Minimizer) for the class of
interest. Our work builds on a recent abstention algorithm of Goldwasser,
Kalais, and Montasser (2020) for transductive binary classification.
