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Finding classifiers robust to adversarial examples is critical for their safedeployment. Determining the robustness of the best possible classifier under agiven threat model for a given data distribution and comparing it to thatachieved by state-of-the-art training methods is thus an important diagnostictool. In this paper, we find achievable information-theoretic lower bounds onloss in the presence of a test-time attacker for multi-class classifiers on anydiscrete dataset. We provide a general framework for finding the optimal 0-1loss that revolves around the construction of a conflict hypergraph from thedata and adversarial constraints. We further define other variants of theattacker-classifier game that determine the range of the optimal loss moreefficiently than the full-fledged hypergraph construction. Our evaluationshows, for the first time, an analysis of the gap to optimal robustness forclassifiers in the multi-class setting on benchmark datasets.

Characterizing the Optimal 0-1 Loss for Multi-class Classification with a Test-time Attacker