Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials
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We study the problem of PAC learning a linear combination of k ReLU activations under the standard Gaussian distribution on ℝ^d with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity (dk/ϵ)^O(k), where ϵ>0 is the target accuracy. Prior work had given an algorithm for this problem with complexity (dk/ϵ)^h(k), where the function h(k) scales super-polynomially in k. Interestingly, the complexity of our algorithm is near-optimal within the class of Correlational Statistical Query algorithms. At a high-level, our algorithm uses tensor decomposition to identify a subspace such that all the O(k)-order moments are small in the orthogonal directions. Its analysis makes essential use of the theory of Schur polynomials to show that the higher-moment error tensors are small given that the lower-order ones are.
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