Research in adversarial learning follows a cat and mouse game between attackers and defenders where attacks are proposed, they are mitigated by new defenses, and subsequently new attacks are proposed that break earlier defenses, and so on. However, it has remained unclear as to whether there are conditions under which no better attacks or defenses can be proposed. In this paper, we propose a game-theoretic framework for studying attacks and defenses which exist in equilibrium. Under a locally linear decision boundary model for the underlying binary classifier, we prove that the Fast Gradient Method attack and a Randomized Smoothing defense form a Nash Equilibrium. We then show how this equilibrium defense can be approximated given finitely many samples from a data-generating distribution, and derive a generalization bound for the performance of our approximation.

The paper provides a game-theoretic analysis of additive attacks in the "No-Box" setting. Its most significant result is the proof that the FGM attack and randomized smoothing form a Nash equilibrium under the assumption of a local linearity of the decision boundary. The paper's main contribution is theoretical, its empirical evaluation is performed on the MNIST dataset for a limited number of classes. Also, the validity of some theoretical assumptions is not convincingly presented in the paper. The authors should also clarify the relationship of their work to prior game-theoretic approaches to adversarial learning, e.g., Brückner, M., Kanzow, C. and Scheffer, T., 2012.

Research in adversarial learning follows a cat and mouse game between attackers and defenders where attacks are proposed, they are mitigated by new defenses, and subsequently new attacks are proposed that break earlier defenses, and so on. However, it has remained unclear as to whether there are conditions under which no better attacks or defenses can be proposed. In this paper, we propose a game-theoretic framework for studying attacks and defenses which exist in equilibrium. Under a locally linear decision boundary model for the underlying binary classifier, we prove that the Fast Gradient Method attack and a Randomized Smoothing defense form a Nash Equilibrium. We then show how this equilibrium defense can be approximated given finitely many samples from a data-generating distribution, and derive a generalization bound for the performance of our approximation.

The main goal of search engines is ad hoc retrieval: ranking documents in a corpus by their relevance to the information need expressed by a query. The Probability Ranking Principle (PRP) --- ranking the documents by their relevance probabilities --- is the theoretical foundation of most existing ad hoc document retrieval methods. A key observation that motivates our work is that the PRP does not account for potential post-ranking effects; specifically, changes to documents that result from a given ranking. Yet, in adversarial retrieval settings such as the Web, authors may consistently try to promote their documents in rankings by changing them. We prove that, indeed, the PRP can be sub-optimal in adversarial retrieval settings. We do so by presenting a novel game theoretic analysis of the adversarial setting. The analysis is performed for different types of documents (single-topic and multi-topic) and is based on different assumptions about the writing qualities of documents' authors. We show that in some cases, introducing randomization into the document ranking function yields an overall user utility that transcends that of applying the PRP.