On Margin Maximization in Linear and ReLU Networks

Gal Vardi, Ohad Shamir, Nathan Srebro

Keywords: Implicit bias, Homogeneous neural networks, Exponential loss, Logistic loss, Maximum margin, Linear networks, ReLU networks

Abstract: The implicit bias of neural networks has been extensively studied in recent years. Lyu and Li [2019] showed that in homogeneous networks trained with the exponential or the logistic loss, gradient flow converges to a KKT point of the max margin problem in the parameter space. However, that leaves open the question of whether this point will generally be an actual optimum of the max margin problem. In this paper, we study this question in detail, for several neural network architectures involving linear and ReLU activations. Perhaps surprisingly, we show that in many cases, the KKT point is not even a local optimum of the max margin problem. On the flip side, we identify multiple settings where a local or global optimum can be guaranteed. Finally, we answer a question posed in Lyu and Li [2019] by showing that for non-homogeneous networks, the normalized margin may strictly decrease over time.

One-sentence Summary: For several architectures of homogeneous neural networks involving linear and ReLU activations, we study whether gradient flow converges to a global/local optimum of the max margin problem in parameter space.