Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials
This provides a near-optimal solution for learning neural networks with ReLU activations, addressing a bottleneck in computational learning theory for practitioners in machine learning.
The paper tackles the problem of PAC learning a one-hidden-layer ReLU network with k activations under Gaussian distribution, achieving an efficient algorithm with sample and computational complexity (dk/ε)^{O(k)}, improving on prior super-polynomial scaling in k.
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 $(dk/ε)^{O(k)}$, 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.