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Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in many practical applications, including point clouds and particle physics, a relevant notion of generalization should include varying the input size. In this work we treat symmetric functions (of any size) as functions over probability measures, and study the learning and representation of neural networks defined on measures. By focusing on shallow architectures, we establish approximation and generalization bounds under different choices of regularization (such as RKHS and variation norms), that capture a hierarchy of functional spaces with increasing degree of non-linear learning. The resulting models can be learned efficiently and enjoy generalization guarantees that extend across input sizes, as we verify empirically. 

Domain adaptation in imitation learning repre- sents an essential step towards improving gen- eralizability. However, even in the restricted setting of third-person imitation where trans- fer is between isomorphic Markov Decision Processes, there are no strong guarantees on the performance of transferred policies. We present problem-dependent, statistical learn- ing guarantees for third-person imitation from observation in an offline setting, and a lower bound on performance in an online setting.