Training Convolutional ReLU Neural Networks in Polynomial Time: Exact Convex Optimization Formulations
We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons and data dimension. Particularly, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several CNN architectures. We first prove that two-layer CNNs can be globally optimized via an $\ell_2$ norm regularized convex program. We then show that certain three-layer CNN training problems are equivalent to an $\ell_1$ regularized convex program. We also extend these results to multi-layer CNN architectures. Furthermore, we present extensions of our approach to different pooling methods.