  We reveal the incoherence between the widely-adopted empirical domain
adversarial training and its generally-assumed theoretical counterpart based on
$\mathcal{H}$-divergence. Concretely, we find that $\mathcal{H}$-divergence is
not equivalent to Jensen-Shannon divergence, the optimization objective in
domain adversarial training. To this end, we establish a new theoretical
framework by directly proving the upper and lower target risk bounds based on
joint distributional Jensen-Shannon divergence. We further derive
bi-directional upper bounds for marginal and conditional shifts. Our framework
exhibits inherent flexibilities for different transfer learning problems, which
is usable for various scenarios where $\mathcal{H}$-divergence-based theory
fails to adapt. From an algorithmic perspective, our theory enables a generic
guideline unifying principles of semantic conditional matching, feature
marginal matching, and label marginal shift correction. We employ algorithms
for each principle and empirically validate the benefits of our framework on
real datasets.
