Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization

TL;DR

This paper analyzes how large stepsize gradient descent optimizes two-layer neural networks with near-homogeneity, showing phase transitions, margin improvements, and efficiency gains over smaller stepsizes, applicable beyond traditional regimes.

Contribution

It provides a theoretical understanding of phase transitions and implicit bias in large stepsize GD for non-homogeneous two-layer networks, extending beyond NTK and mean-field regimes.

Findings

Second phase begins after risk drops below a threshold.

Normalized margin grows nearly monotonically in the second phase.

Large stepsize GD is more efficient than small stepsize GD.

Abstract

The typical training of neural networks using large stepsize gradient descent (GD) under the logistic loss often involves two distinct phases, where the empirical risk oscillates in the first phase but decreases monotonically in the second phase. We investigate this phenomenon in two-layer networks that satisfy a near-homogeneity condition. We show that the second phase begins once the empirical risk falls below a certain threshold, dependent on the stepsize. Additionally, we show that the normalized margin grows nearly monotonically in the second phase, demonstrating an implicit bias of GD in training non-homogeneous predictors. If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps. Finally, we demonstrate that by choosing a suitably large stepsize, GD that undergoes this phase transition is more efficient than GD that monotonically decreases the risk. Our analysis applies to networks of any width, beyond the well-known neural tangent kernel and mean-field regimes.