Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials

• URL: http://arxiv.org/abs/2307.12840v2
• Date: Tue, 25 Jul 2023 17:25:03 GMT
• Title: Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials
• Authors: Ilias Diakonikolas and Daniel M. Kane
• Abstract summary: We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
• Score: 50.90125395570797
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $\mathbb{R}^d$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/\epsilon)^{O(k)}$, where $\epsilon>0$ is the target accuracy. Prior work had given an algorithm for this problem with complexity $(dk/\epsilon)^{h(k)}$, 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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