PATH REGULARIZATION: A CONVEXITY AND SPAR-SITY INDUCING REGULARIZATION FOR PARALLEL RELU NETWORKS

Abstract

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)

1. INTRODUCTION

Deep Neural Networks (DNNs) have achieved substantial improvements in several fields of machine learning. However, since DNNs have a highly nonlinear and non-convex structure, the fundamental principles behind their remarkable performance is still an open problem. Therefore, advances in this field largely depend on heuristic approaches. One of the most prominent techniques to boost the generalization performance of DNNs is regularizing layer weights so that the network can fit a function that performs well on unseen test data. Even though weight decay, i.e., penalizing the 2 2norm of the layer weights, is commonly employed as a regularization technique in practice, recently, it has been shown that 2 -path regularizer (Neyshabur et al., 2015b) , i.e., the sum over all paths in the network of the squared product over all weights in the path, achieves further empirical gains (Neyshabur et al., 2015a) . Therefore, in this paper, we investigate the underlying mechanisms behind path regularized DNNs through the lens of convex optimization.



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).

