  Recent advances in domain adaptation establish that requiring a low risk on
the source domain and equal feature marginals degrade the adaptation's
performance. At the same time, empirical evidence shows that incorporating an
unsupervised target domain term that pushes decision boundaries away from the
high-density regions, along with relaxed alignment, improves adaptation. In
this paper, we theoretically justify such observations via a new bound on the
target risk, and we connect two notions of relaxation for divergence, namely
$\beta-$relaxed divergences and localization. This connection allows us to
incorporate the source domain's categorical structure into the relaxation of
the considered divergence, provably resulting in a better handling of the label
shift case in particular.
