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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.