  Understanding the fundamental mechanism behind the success of deep neural
networks is one of the key challenges in the modern machine learning
literature. Despite numerous attempts, a solid theoretical analysis is yet to
be developed. In this paper, we develop a novel unified framework to reveal a
hidden regularization mechanism through the lens of convex optimization. We
first show that the training of multiple three-layer ReLU sub-networks with
weight decay regularization can be equivalently cast as a convex optimization
problem in a higher dimensional space, where sparsity is enforced via a group
$\ell_1$-norm regularization. Consequently, ReLU networks can be interpreted as
high dimensional feature selection methods. More importantly, we then prove
that the equivalent convex problem can be globally optimized by a standard
convex optimization solver with a polynomial-time complexity with respect to
the number of samples and data dimension when the width of the network is
fixed. Finally, we numerically validate our theoretical results via experiments
involving both synthetic and real datasets.
