We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the nonconvex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the nonconvex training objective. Since all stationary points of the ReLU training problem can be represented as optima of subsampled convex programs, our work provides a general expression for all critical points of the nonconvex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks.
more »
« less
Optimal Sets and Solution Paths of ReLU Networks
We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the nonconvex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the nonconvex training objective. Since all stationary points of the ReLU training problem can be represented as optima of subsampled convex programs, our work provides a general expression for all critical points of the nonconvex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks.
more »
« less
 NSFPAR ID:
 10433860
 Date Published:
 Journal Name:
 International Conference on Machine Learning
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


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.more » « less

null (Ed.)We develop a convex analytic framework for ReLU neural networks which elucidates the inner workings of hidden neurons and their function space characteristics. We show that neural networks with rectified linear units act as convex regularizers, where simple solutions are encouraged via extreme points of a certain convex set. For one dimensional regression and classification, as well as rankone data matrices, we prove that finite twolayer ReLU networks with norm regularization yield linear spline interpolation. We characterize the classification decision regions in terms of a closed form kernel matrix and minimum L1 norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain neural network predictions with finitely many neurons. Our convex geometric description also provides intuitive explanations of hidden neurons as auto encoders. In higher dimensions, we show that the training problem for twolayer networks can be cast as a finite dimensional convex optimization problem with infinitely many constraints. We then provide a family of convex relaxations to approximate the solution, and a cuttingplane algorithm to improve the relaxations. We derive conditions for the exactness of the relaxations and provide simple closed form formulas for the optimal neural network weights in certain cases. We also establish a connection to ℓ0ℓ1 equivalence for neural networks analogous to the minimal cardinality solutions in compressed sensing. Extensive experimental results show that the proposed approach yields interpretable and accurate models.more » « less

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 corresponds with the solution of Stochastic Gradient Descent in practice.more » « less

We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for onedimensional datasets which was previously known only for twolayer networks. Furthermore, we provide closedform solutions for the optimal layer weights when data is rankone or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.more » « less