Implicit Convex Regularizers of CNN Architectures: Convex Optimization of Two- and Three-Layer Networks in Polynomial Time
This provides a theoretical foundation for globally optimizing CNN training, which could benefit researchers and practitioners in machine learning by offering efficient and interpretable solutions.
The authors tackled the non-convex optimization problem in training Convolutional Neural Networks (CNNs) by introducing exact convex optimization formulations for two- and three-layer CNNs with ReLU activations, achieving polynomial-time complexity with respect to data samples, neurons, and dimension.
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. More specifically, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several two- and three-layer 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 multi-layer circular CNN training problems with a single ReLU layer are equivalent to an $\ell_1$ regularized convex program that encourages sparsity in the spectral domain. We also extend these results to three-layer CNNs with two ReLU layers. Furthermore, we present extensions of our approach to different pooling methods, which elucidates the implicit architectural bias as convex regularizers.