On Margin Maximization in Linear and ReLU Networks
This addresses a theoretical gap in understanding implicit bias for researchers in machine learning theory, but is incremental as it builds on prior work by Lyu and Li.
The paper investigates whether gradient flow in homogeneous neural networks converges to an optimum of the max margin problem, finding that in many cases with linear and ReLU activations, it does not reach even a local optimum, but identifies specific settings where optimality can be guaranteed.
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.