The performance of ML models degrades when the training population is different from that seen under operation. Towards assessing distributional robustness, we study the worst-case performance of a model over all subpopulations of a given size, defined with respect to core attributes $Z$. This notion of robustness can consider arbitrary (continuous) attributes $Z$, and automatically accounts for complex intersectionality in disadvantaged groups. We develop a scalable yet principled two-stage estimation procedure that can evaluate the robustness of state-of-the-art models. We prove that our procedure enjoys several finite-sample convergence guarantees, including dimension-free convergence. Instead of overly conservative notions based on Rademacher complexities, our evaluation error depends on the dimension of $Z$ only through the out-of-sample error in estimating the performance conditional on $Z$. On real datasets, we demonstrate that our method certifies the robustness of a model and prevents deployment of unreliable models.