Title: When is a Convolutional Filter Easy To Learn?,
Abstract: We analyze the convergence of (stochastic) gradient descent algorithm for
learning a convolutional filter with Rectified Linear Unit (ReLU) activation
function. Our analysis does not rely on any specific form of the input
distribution and our proofs only use the definition of ReLU, in contrast with
previous works that are restricted to standard Gaussian input. We show that
(stochastic) gradient descent with random initialization can learn the
convolutional filter in polynomial time and the convergence rate depends on the
smoothness of the input distribution and the closeness of patches. To the best
of our knowledge, this is the first recovery guarantee of gradient-based
algorithms for convolutional filter on non-Gaussian input distributions. Our
theory also justifies the two-stage learning rate strategy in deep neural
networks. While our focus is theoretical, we also present experiments that
illustrate our theoretical findings.