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Counterfactual (CF) explanations for machine learning (ML) models are preferred by end-users, as they explain the predictions of ML models by providing a recourse (or contrastive) case to individuals who are adversely impacted by predicted outcomes. Existing CF explanation methods generate recourses under the assumption that the underlying target ML model remains stationary over time. However, due to commonly occurring distributional shifts in training data, ML models constantly get updated in practice, which might render previously generated recourses invalid and diminish end-users trust in our algorithmic framework. To address this problem, we propose RoCourseNet, a training framework that jointly optimizes predictions and recourses that are robust to future data shifts. This work contains four key contributions: (1) We formulate the robust recourse generation problem as a tri-level optimization problem which consists of two sub-problems: (i) a bi-level problem that finds the worst-case adversarial shift in the training data, and (ii) an outer minimization problem to generate robust recourses against this worst-case shift. (2) We leverage adversarial training to solve this tri-level optimization problem by: (i) proposing a novel virtual data shift (VDS) algorithm to find worst-case shifted ML models via explicitly considering the worst-case data shift in the training dataset, and (ii) a block-wise coordinate descent procedure to optimize for prediction and corresponding robust recourses. (3) We evaluate RoCourseNet's performance on three real-world datasets, and show that RoCourseNet consistently achieves more than 96% robust validity and outperforms state-of-the-art baselines by at least 10% in generating robust CF explanations. (4) Finally, we generalize the RoCourseNet framework to accommodate any parametric post-hoc methods for improving robust validity.