Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization
This work addresses the optimization efficiency and implicit bias of gradient descent in neural network training, providing theoretical insights for machine learning researchers, though it is incremental as it builds on known phase transition phenomena.
The paper investigates the two-phase behavior of large stepsize gradient descent (GD) in training non-homogeneous two-layer neural networks under the logistic loss, showing that the second phase begins when empirical risk drops below a stepsize-dependent threshold and leads to nearly monotonic growth in normalized margin, with GD being more efficient than monotonic risk-decreasing GD when using a suitably large stepsize.
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.