  As learning solutions reach critical applications in social, industrial, and
medical domains, the need to curtail their behavior has become paramount. There
is now ample evidence that without explicit tailoring, learning can lead to
biased, unsafe, and prejudiced solutions. To tackle these problems, we develop
a generalization theory of constrained learning based on the probably
approximately correct (PAC) learning framework. In particular, we show that
imposing requirements does not make a learning problem harder in the sense that
any PAC learnable class is also PAC constrained learnable using a constrained
counterpart of the empirical risk minimization (ERM) rule. For typical
parametrized models, however, this learner involves solving a constrained
non-convex optimization program for which even obtaining a feasible solution is
challenging. To overcome this issue, we prove that under mild conditions the
empirical dual problem of constrained learning is also a PAC constrained
learner that now leads to a practical constrained learning algorithm based
solely on solving unconstrained problems. We analyze the generalization
properties of this solution and use it to illustrate how constrained learning
can address problems in fair and robust classification.
