Efficiently Learning One Hidden Layer ReLU Networks From Queries

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/ ) O(k) , where > 0 is the target accuracy. Prior work had given an algorithm for this problem with complexity (dk/ ) 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.