Sample Complexity for Distributionally Robust Learning under ²-divergence

This paper investigates the sample complexity of learning a distributionally robust predictor under a particular distributional shift based on chi(2)-divergence, which is well known for its computational feasibility and statistical properties. We demonstrate that any hypothesis class H with finite VC dimension is distributionally robustly learnable. Moreover, we show that when the perturbation size is smaller than a constant, finite VC dimension is also necessary for distributionally robust learning by deriving a lower bound of sample complexity in terms of VC dimension.