Abstract: 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 fixed 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 onrobust loss in the presence of a test-time attacker for *multi-classclassifiers on any discrete dataset*. We provide a general framework for findingthe optimal $0-1$ loss that revolves around the construction of a conflicthypergraph from the data and adversarial constraints. The prohibitive cost ofthis formulation in practice leads us to formulate other variants of the attacker-classifiergame that more efficiently determine the range of the optimal loss. Ourvaluation shows, for the first time, an analysis of the gap to optimalrobustness for classifiers in the multi-class setting on benchmark datasets.