Convex Duality of Deep Neural Networks

arXiv (Cornell University), 2021

Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature. To this end, we study the training problem of deep neural networks and introduce an analytic approach to unveil hidden convexity in the optimization landscape. We consider a deep parallel ReLU network architecture, which also includes standard deep networks and ResNets as its special cases. We then show that pathwise regularized training problems can be represented as an exact convex optimization problem. We further prove that the equivalent convex problem is regularized via a group sparsity inducing norm. Thus, a path regularized parallel ReLU network can be viewed as a parsimonious convex model in high dimensions. More importantly, since the original training problem may not be trainable in polynomial-time, we propose an approximate algorithm with a fully polynomial-time complexity in all data dimensions. Then, we prove strong global optimality guarantees for this algorithm. We also provide experiments corroborating our theory. (a) 2-layer NN with WD (b) 3-layer NN with WD (c) 3-layer NN with PR (Ours) Figure 1: Decision boundaries of 2-layer and 3-layer ReLU networks that are globally optimized with weight decay (WD) and path regularization (PR). Here, our convex training approach in (c) successfully learns the underlying spiral pattern for each class while the previously studied convex models in (a) and (b) fail (see Appendix A.1 for details).

ArXiv, 2021

We describe the convex semi-infinite dual of the two-layer vector-output ReLU neural network training problem. This semi-infinite 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 non-convex neural network training problem is equivalent to a finite-dimensional 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 semi-nonnegative 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...

arXiv (Cornell University), 2020

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 2 norm regularized convex program. We then show that multi-layer 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 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.

ArXiv, 2020

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. Particularly, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several 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 certain three-layer CNN training problems are equivalent to an $\ell_1$ regularized convex program. We also extend these results to multi-layer CNN architectures. Furthermore, we present extensions of our approach to different pooling methods.

Advances in neural …, 2006

Convexity has recently received a lot of attention in the machine learning community, and the lack of convexity has been seen as a major disadvantage of many learning algorithms, such as multi-layer artificial neural networks. We show that training multi-layer neural networks in which the number of hidden units is learned can be viewed as a convex optimization problem. This problem involves an infinite number of variables, but can be solved by incrementally inserting a hidden unit at a time, each time finding a linear classifier that minimizes a weighted sum of errors. s(a) = 1 1+e −a . A learning algorithm must specify how to select m, the w i 's and the v i 's.

2019

We consider non-convex training of shallow neural networks and introduce a convex relaxation approach with theoretical guarantees. For the single neuron case, we prove that the relaxation preserves the location of the global minimum under a planted model assumption. Therefore, a globally optimal solution can be efficiently found via a gradient method. We show that gradient descent applied on the relaxation always outperforms gradient descent on the original non-convex loss with no additional computational cost. We then characterize this relaxation as a regularizer and further introduce extensions to multineuron single hidden layer networks.

Cornell University - arXiv, 2021

Batch Normalization (BN) is a commonly used technique to accelerate and stabilize training of deep neural networks. Despite its empirical success, a full theoretical understanding of BN is yet to be developed. In this work, we analyze BN through the lens of convex optimization. We introduce an analytic framework based on convex duality to obtain exact convex representations of weight-decay regularized ReLU networks with BN, which can be trained in polynomial-time. Our analyses also show that optimal layer weights can be obtained as simple closed-form formulas in the high-dimensional and/or overparameterized regimes. Furthermore, we find that Gradient Descent provides an algorithmic bias effect on the standard non-convex BN network, and we design an approach to explicitly encode this implicit regularization into the convex objective. Experiments with CIFAR image classification highlight the effectiveness of this explicit regularization for mimicking and substantially improving the performance of standard BN networks.

2020

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 rectified linear units in neural networks act as convex regularizers, where simple solutions are encouraged via extreme points of a certain convex set. For one dimensional regression and classification, we prove that finite two-layer ReLU networks with norm regularization yield linear spline interpolation. In the more general higher dimensional case, we show that the training problem for twolayer networks can be cast as a convex optimization problem with infinitely many constraints. We then provide a family of convex relaxations to approximate the solution, and a cutting-plane 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. Our results show that the hidden neurons of a R...

2022

We develop fast algorithms and robust software for convex optimization of two-layer neural networks with ReLU activation functions. Our work leverages a convex reformulation of the standard weight-decay penalized training problem as a set of group-$\ell_1$-regularized data-local models, where locality is enforced by polyhedral cone constraints. In the special case of zero-regularization, we show that this problem is exactly equivalent to unconstrained optimization of a convex "gated ReLU" network. For problems with non-zero regularization, we show that convex gated ReLU models obtain data-dependent approximation bounds for the ReLU training problem. To optimize the convex reformulations, we develop an accelerated proximal gradient method and a practical augmented Lagrangian solver. We show that these approaches are faster than standard training heuristics for the non-convex problem, such as SGD, and outperform commercial interior-point solvers. Experimentally, we verify ou...

2019

We develop a convex analytic framework for ReLU networks and study their function space characteristics. We show that the optimal parameters of two layer over-parameterized networks can be described as extreme points of a certain convex set. For one dimensional regression, our theory explains why ReLU networks yield linear spline interpolation. In higher dimensions, we show that the training problem can be cast as a convex optimization problem with infinitely many constraints. We then provide convex relaxations to approximate the solution, and a cutting-plane algorithm to improve the relaxations.