Implicit Bias of Gradient Descent for Non-Homogeneous Deep Networks

TL;DR

This paper characterizes the asymptotic behavior of gradient descent in non-homogeneous deep networks, showing convergence properties and margin maximization conditions, extending understanding beyond homogeneous networks.

Contribution

It provides the first analysis of the implicit bias of gradient descent in non-homogeneous deep networks, including residual connections and non-homogeneous activations.

Findings

Normalized margin increases nearly monotonically

GD iterates' direction converges despite diverging norm

Limit points satisfy KKT conditions of margin maximization

Abstract

We establish the asymptotic implicit bias of gradient descent (GD) for generic non-homogeneous deep networks under exponential loss. Specifically, we characterize three key properties of GD iterates starting from a sufficiently small empirical risk, where the threshold is determined by a measure of the network's non-homogeneity. First, we show that a normalized margin induced by the GD iterates increases nearly monotonically. Second, we prove that while the norm of the GD iterates diverges to infinity, the iterates themselves converge in direction. Finally, we establish that this directional limit satisfies the Karush-Kuhn-Tucker (KKT) conditions of a margin maximization problem. Prior works on implicit bias have focused exclusively on homogeneous networks; in contrast, our results apply to a broad class of non-homogeneous networks satisfying a mild near-homogeneity condition. In particular, our results apply to networks with residual connections and non-homogeneous activation functions, thereby resolving an open problem posed by Ji and Telgarsky (2020).