On Robust Multiclass Learnability

This work analyzes the robust learning problem in the multiclass setting. Under the framework of Probably Approximately Correct (PAC) learning, we first show that the graph dimension and the Natarajan dimension, which characterize the standard multiclass learnability, are no longer applicable in robust learning problem. We then generalize these notions to the robust learning setting, denoted as the adversarial graph dimension (AG-dimension) and the adversarial Natarajan dimension (AN-dimension). Upper and lower bounds of the sample complexity of robust multiclass learning are rigorously derived based on the AG-dimension and AN-dimension, respectively. Moreover, we calculate the AG-dimension and AN-dimension of the class of linear multiclass predictors, and show that the graph (Natarajan) dimension is of the same order as the AG(AN)-dimension. Finally, we prove that the AG-dimension and AN-dimension are not equivalent.