  Unsupervised Domain Adaptation (UDA) aims to bridge the gap between a source
domain, where labelled data are available, and a target domain only represented
with unlabelled data. If domain invariant representations have dramatically
improved the adaptability of models, to guarantee their good transferability
remains a challenging problem. This paper addresses this problem by using
active learning to annotate a small budget of target data. Although this setup,
called Active Domain Adaptation (ADA), deviates from UDA's standard setup, a
wide range of practical applications are faced with this situation. To this
purpose, we introduce \textit{Stochastic Adversarial Gradient Embedding}
(SAGE), a framework that makes a triple contribution to ADA. First, we select
for annotation target samples that are likely to improve the representations'
transferability by measuring the variation, before and after annotation, of the
transferability loss gradient. Second, we increase sampling diversity by
promoting different gradient directions. Third, we introduce a novel training
procedure for actively incorporating target samples when learning invariant
representations. SAGE is based on solid theoretical ground and validated on
various UDA benchmarks against several baselines. Our empirical investigation
demonstrates that SAGE takes the best of uncertainty \textit{vs} diversity
samplings and improves representations transferability substantially.
