  We study the sample complexity of learning one-hidden-layer convolutional
neural networks (CNNs) with non-overlapping filters. We propose a novel
algorithm called approximate gradient descent for training CNNs, and show that,
with high probability, the proposed algorithm with random initialization grants
a linear convergence to the ground-truth parameters up to statistical
precision. Compared with existing work, our result applies to general
non-trivial, monotonic and Lipschitz continuous activation functions including
ReLU, Leaky ReLU, Sigmod and Softplus etc. Moreover, our sample complexity
beats existing results in the dependency of the number of hidden nodes and
filter size. In fact, our result matches the information-theoretic lower bound
for learning one-hidden-layer CNNs with linear activation functions, suggesting
that our sample complexity is tight. Our theoretical analysis is backed up by
numerical experiments.
