Implicit Convex Regularizers of CNN Architectures: Convex Optimization of Two- and Three-Layer Networks in Polynomial Time

• URL: http://arxiv.org/abs/2006.14798v3
• Date: Thu, 18 Mar 2021 15:30:26 GMT
• Title: Implicit Convex Regularizers of CNN Architectures: Convex Optimization of Two- and Three-Layer Networks in Polynomial Time
• Authors: Tolga Ergen, Mert Pilanci
• Abstract summary: We study training of Convolutional Neural Networks (CNNs) with ReLU activations. We introduce exact convex optimization with a complexity with respect to the number of data samples, the number of neurons, and data dimension.
• Score: 70.15611146583068
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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