Title: Probabilistic Categorical Adversarial Attack and Adversarial Training. Abstract: The existence of adversarial examples brings huge concern for people to apply Deep Neural Networks (DNNs) in safety-critical tasks. However, how to generate adversarial examples with categorical data is an important problem but lacks extensive exploration. Previously established methods leverage greedy search methods, which can be very time-consuming to conduct a successful attack. This also limits the development of adversarial training and potential defenses for categorical data. To tackle this problem, we propose a Probabilistic Categorical Adversarial Attack (PCAA), which transfers the discrete optimization problem to a continuous problem that can be solved efficiently by Projected Gradient Descent. In our paper, we theoretically analyze its optimality and time complexity to demonstrate its significant advantage over current greedy-based attacks. Moreover, based on our attack, we propose an efficient adversarial training framework. Through a comprehensive empirical study, we justify the effectiveness of our proposed attack and defense algorithms. 