We develop a convex analytic approach to analyze finite width twolayer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rankone data. We also characterize the classification decision regions in terms of a kernel matrix and minimum `1norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as autoencoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cuttingplane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closedform formulas in some practically relevant special cases.
Vectoroutput ReLU Neural Network Problems are Copositive Programs: Convex Analysis of Two Layer Networks and Polynomialtime Algorithms
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 which more »
 Award ID(s):
 1838179
 Publication Date:
 NSFPAR ID:
 10310557
 Journal Name:
 International Conference on Learnining Representations (ICLR)
 Sponsoring Org:
 National Science Foundation
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