Although deep convolutional neural networks (CNNs) have demonstrated remarkable performance on multiple computer vision tasks, researches on adversarial learning have shown that deep models are vulnerable to adversarial examples, which are crafted by adding visually imperceptible perturbations to the input images. Most of the existing adversarial attack methods only create a single adversarial example for the input, which just gives a glimpse of the underlying data manifold of adversarial examples. An attractive solution is to explore the solution space of the adversarial examples and generate a diverse bunch of them, which could potentially improve the robustness of real-world systems and help prevent severe security threats and vulnerabilities. In this paper, we present an effective method, called Hamiltonian Monte Carlo with Accumulated Momentum (HMCAM), aiming to generate a sequence of adversarial examples. To improve the efficiency of HMC, we propose a new regime to automatically control the length of trajectories, which allows the algorithm to move with adaptive step sizes along the search direction at different positions. Moreover, we revisit the reason for high computational cost of adversarial training under the view of MCMC and design a new generative method called Contrastive Adversarial Training (CAT), which approaches equilibrium distribution of adversarial examples with only few iterations by building from small modifications of the standard Contrastive Divergence (CD) and achieve a trade-off between efficiency and accuracy. Both quantitative and qualitative analysis on several natural image datasets and practical systems have confirmed the superiority of the proposed algorithm.
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