  Sparsification of neural networks is one of the effective complexity
reduction methods to improve efficiency and generalizability. We consider the
problem of learning a one hidden layer convolutional neural network with ReLU
activation function via gradient descent under sparsity promoting penalties. It
is known that when the input data is Gaussian distributed, no-overlap networks
(without penalties) in regression problems with ground truth can be learned in
polynomial time at high probability. We propose a relaxed variable splitting
method integrating thresholding and gradient descent to overcome the lack of
non-smoothness in the loss function. The sparsity in network weight is realized
during the optimization (training) process. We prove that under $\ell_1,
\ell_0$; and transformed-$\ell_1$ penalties, no-overlap networks can be learned
with high probability, and the iterative weights converge to a global limit
which is a transformation of the true weight under a novel thresholding
operation. Numerical experiments confirm theoretical findings, and compare the
accuracy and sparsity trade-off among the penalties.
