We describe the convex semiinfinite dual of the twolayer vectoroutput ReLU neural network training problem. This semiinfinite dual admits a finite dimensional representation, but its support is over a convex set which is difficult to characterize. In particular, we demonstrate that the nonconvex neural network training problem is equivalent to a finitedimensional convex copositive program. Our work is the first to identify this strong connection between the global optima of neural networks and those of copositive programs. We thus demonstrate how neural networks implicitly attempt to solve copositive programs via seminonnegative matrix factorization, and draw key insights from this formulation. We describe the first algorithms for provably finding the global minimum of the vector output neural network training problem, which are polynomial in the number of samples for a fixed data rank, yet exponential in the dimension. However, in the case of convolutional architectures, the computational complexity is exponential in only the filter size and polynomial in all other parameters. We describe the circumstances in which we can find the global optimum of this neural network training problem exactly with softthresholded SVD, and provide a copositive relaxation which is guaranteed to be exact for certain classes of problems, and whichmore »
Implicit Convex Regularizers of CNN Architectures: Convex Optimization of Two and ThreeLayer Networks in Polynomial Time
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 semiinfinite duality to obtain equivalent convex optimization problems for several two and threelayer CNN architectures. We first prove that twolayer CNNs can be globally optimized via an `2 norm regularized convex program. We then show that multilayer circular CNN training problems with a single ReLU layer are equivalent to an `1 regularized convex program that encourages sparsity in the spectral domain. We also extend these results to threelayer 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.
 Award ID(s):
 1838179
 Publication Date:
 NSFPAR ID:
 10310558
 Journal Name:
 International Conference on Learning Representations (ICLR) 2021
 Sponsoring Org:
 National Science Foundation
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