  The vast majority of existing algorithms for unsupervised domain adaptation
(UDA) focus on adapting from a labeled source domain to an unlabeled target
domain directly in a one-off way. Gradual domain adaptation (GDA), on the other
hand, assumes a path of $(T-1)$ unlabeled intermediate domains bridging the
source and target, and aims to provide better generalization in the target
domain by leveraging the intermediate ones. Under certain assumptions, Kumar et
al. (2020) proposed a simple algorithm, Gradual Self-Training, along with a
generalization bound in the order of $e^{O(T)}
\left(\varepsilon_0+O\left(\sqrt{log(T)/n}\right)\right)$ for the target domain
error, where $\varepsilon_0$ is the source domain error and $n$ is the data
size of each domain. Due to the exponential factor, this upper bound becomes
vacuous when $T$ is only moderately large. In this work, we analyze gradual
self-training under more general and relaxed assumptions, and prove a
significantly improved generalization bound as $\varepsilon_0+ O \left(T\Delta
+ T/\sqrt{n}\right) + \widetilde{O}\left(1/\sqrt{nT}\right)$, where $\Delta$ is
the average distributional distance between consecutive domains. Compared with
the existing bound with an exponential dependency on $T$ as a multiplicative
factor, our bound only depends on $T$ linearly and additively. Perhaps more
interestingly, our result implies the existence of an optimal choice of $T$
that minimizes the generalization error, and it also naturally suggests an
optimal way to construct the path of intermediate domains so as to minimize the
accumulative path length $T\Delta$ between the source and target. To
corroborate the implications of our theory, we examine gradual self-training on
multiple semi-synthetic and real datasets, which confirms our findings. We
believe our insights provide a path forward toward the design of future GDA
algorithms.
