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Stochastic Security: Adversarial Defense Using Long-Run Dynamics of Energy-Based Models

Mitchell Hill · Jonathan Mitchell · Song-Chun Zhu

Keywords: [ Langevin sampling ] [ Markov chain Monte Carlo ] [ Energy-based model ] [ adversarial defense ] [ adversarial robustness ] [ adversarial attack ]


The vulnerability of deep networks to adversarial attacks is a central problem for deep learning from the perspective of both cognition and security. The current most successful defense method is to train a classifier using adversarial images created during learning. Another defense approach involves transformation or purification of the original input to remove adversarial signals before the image is classified. We focus on defending naturally-trained classifiers using Markov Chain Monte Carlo (MCMC) sampling with an Energy-Based Model (EBM) for adversarial purification. In contrast to adversarial training, our approach is intended to secure highly vulnerable pre-existing classifiers. To our knowledge, no prior defensive transformation is capable of securing naturally-trained classifiers, and our method is the first to validate a post-training defense approach that is distinct from current successful defenses which modify classifier training.

The memoryless behavior of long-run MCMC sampling will eventually remove adversarial signals, while metastable behavior preserves consistent appearance of MCMC samples after many steps to allow accurate long-run prediction. Balancing these factors can lead to effective purification and robust classification. We evaluate adversarial defense with an EBM using the strongest known attacks against purification. Our contributions are 1) an improved method for training EBM's with realistic long-run MCMC samples for effective purification, 2) an Expectation-Over-Transformation (EOT) defense that resolves ambiguities for evaluating stochastic defenses and from which the EOT attack naturally follows, and 3) state-of-the-art adversarial defense for naturally-trained classifiers and competitive defense compared to adversarial training on CIFAR-10, SVHN, and CIFAR-100. Our code and pre-trained models are available at

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