A Non-local Convergence Analysis of Gradient Flow for Deep Linear Networks

Area of Science:

• Deep Learning Theory
• Neural Network Optimization
• Mathematical Analysis

Background:

• Deep linear networks are a simplified model for understanding neural network training dynamics.
• Existing analyses often focus on the 'lazy training' regime, limiting insights into broader convergence behaviors.
• Understanding convergence properties is crucial for designing effective deep learning models.

Purpose of the Study:

• To investigate the non-local convergence properties of deep linear networks with at least one single-neuron layer.
• To analyze the behavior of gradient flow trajectories, including convergence to saddle points.
• To determine explicit convergence rates for these trajectories under quadratic loss.

Main Methods:

• Mathematical analysis of gradient flow dynamics.
• Study of deep linear networks with specific architectural constraints (one-neuron layer).
• Analysis under quadratic loss function and arbitrary balanced initialization.

Main Results:

• Characterization of convergent points for gradient flow trajectories, including saddle points.
• Identification of stage-wise convergence rates, ranging from sublinear to linear.
• Demonstration of non-local convergence properties for deep linear networks.

Conclusions:

• The study provides the first explicit non-local analysis of deep linear networks with arbitrary balanced initialization under quadratic loss.
• Findings extend beyond the prevalent lazy training regime, offering a more comprehensive understanding.
• The results contribute to the theoretical foundation of deep learning optimization.