Implicit Bias of Gradient Descent for Non-Homogeneous Deep Networks
This resolves an open problem in understanding optimization dynamics for practical deep networks with non-homogeneous components, such as residual connections, which is incremental but important for theoretical ML.
The authors characterized the asymptotic implicit bias of gradient descent for non-homogeneous deep networks under exponential loss, showing that GD iterates converge in direction to a solution satisfying KKT conditions for margin maximization, extending prior work limited to homogeneous networks.
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).