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Unsupervised domain adaption (UDA) is a transfer learning task where the data and annotations of the source domain are available but only have access to the unlabeled target data during training. Most previous methods try to minimise the domain gap by performing distribution alignment between the source and target domains, which has a notable limitation, i.e., operating at the domain level, but neglecting the sample-level differences. To mitigate this weakness, we propose to improve the unsupervised domain adaptation task with an inter-domain sample matching scheme. We apply the widely-used and robust Triplet loss to match the inter-domain samples. To reduce the catastrophic effect of the inaccurate pseudo-labels generated during training, we propose a novel uncertainty measurement method to select reliable pseudo-labels automatically and progressively refine them. We apply the advanced discrete relaxation Gumbel Softmax technique to realise an adaptive Topk scheme to fulfil the functionality. In addition, to enable the global ranking optimisation within one batch for the domain matching, the whole model is optimised via a novel reinforced attention mechanism with supervision from the policy gradient algorithm, using the Average Precision (AP) as the reward. Our model (termed \textbf{\textit{AdaTriplet-RA}}) achieves State-of-the-art results on several public benchmark datasets, and its effectiveness is validated via comprehensive ablation studies. Our method improves the accuracy of the baseline by 9.7\% (ResNet-101) and 6.2\% (ResNet-50) on the VisDa dataset and 4.22\% (ResNet-50) on the Domainnet dataset. {The source code is publicly available at \textit{https://github.com/shuxy0120/AdaTriplet-RA}}.