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Time series anomaly detection (TSAD) is an important data mining task with numerous applications in the IoT era. In recent years, a large number of deep neural network-based methods have been proposed, demonstrating significantly better performance than conventional methods on addressing challenging TSAD problems in a variety of areas. Nevertheless, these deep TSAD methods typically rely on a clean training dataset that is not polluted by anomalies to learn the "normal profile" of the underlying dynamics. This requirement is nontrivial since a clean dataset can hardly be provided in practice. Moreover, without the awareness of their robustness, blindly applying deep TSAD methods with potentially contaminated training data can possibly incur significant performance degradation in the detection phase. In this work, to tackle this important challenge, we firstly investigate the robustness of commonly used deep TSAD methods with contaminated training data which provides a guideline for applying these methods when the provided training data are not guaranteed to be anomaly-free. Furthermore, we propose a model-agnostic method which can effectively improve the robustness of learning mainstream deep TSAD models with potentially contaminated data. Experiment results show that our method can consistently prevent or mitigate performance degradation of mainstream deep TSAD models on widely used benchmark datasets.