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Convergence of a Relaxed Variable Splitting Method for Learning Sparse Neural Networks via $\ell_1, \ell_0$, and transformed-$\ell_1$ Penalties

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.