  A key problem in a variety of applications is that of domain adaptation from
a public source domain, for which a relatively large amount of labeled data
with no privacy constraints is at one's disposal, to a private target domain,
for which a private sample is available with very few or no labeled data. In
regression problems with no privacy constraints on the source or target data, a
discrepancy minimization algorithm based on several theoretical guarantees was
shown to outperform a number of other adaptation algorithm baselines. Building
on that approach, we design differentially private discrepancy-based algorithms
for adaptation from a source domain with public labeled data to a target domain
with unlabeled private data. The design and analysis of our private algorithms
critically hinge upon several key properties we prove for a smooth
approximation of the weighted discrepancy, such as its smoothness with respect
to the $\ell_1$-norm and the sensitivity of its gradient. Our solutions are
based on private variants of Frank-Wolfe and Mirror-Descent algorithms. We show
that our adaptation algorithms benefit from strong generalization and privacy
guarantees and report the results of experiments demonstrating their
effectiveness.
