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In this article, we study the problem of robust reconfigurable intelligent surface (RIS)-aided downlink communication over heterogeneous RIS types in the supervised learning setting. By modeling downlink communication over heterogeneous RIS designs as different workers that learn how to optimize phase configurations in a distributed manner, we solve this distributed learning problem using a distributionally robust formulation in a communication-efficient manner, while establishing its rate of convergence. By doing so, we ensure that the global model performance of the worst-case worker is close to the performance of other workers. Simulation results show that our proposed algorithm requires fewer communication rounds (about 50% lesser) to achieve the same worst-case distribution test accuracy compared to competitive baselines.