Generalization Guarantees of Gradient Descent for Shallow Neural Networks

Area of Science:

• Machine Learning
• Deep Learning Theory
• Algorithmic Stability

Background:

• Understanding neural network (NN) generalization is crucial for reliable AI.
• Algorithmic stability provides a framework for analyzing generalization.
• Previous studies primarily focused on single-hidden-layer networks, neglecting network scaling effects.

Purpose of the Study:

• To extend algorithmic stability and generalization analysis to two- and three-layer neural networks trained by gradient descent (GD).
• To investigate the impact of network scaling on generalization.
• To derive conditions for achieving optimal risk rates in NNs.

Main Methods:

• Comprehensive stability and generalization analysis of GD for two- and three-layer NNs.
• Relaxing previous conditions for two-layer NNs under general network scaling.
• Utilizing a novel induction strategy to demonstrate the nearly co-coercive property of three-layer NNs, considering overparameterization.

Main Results:

• Derived an excess risk rate of O(1/n) for GD in both two- and three-layer NNs.
• Identified sufficient and necessary conditions for under- and over-parameterized NNs to achieve the O(1/n) risk rate.
• Demonstrated that increased scaling factors or decreased network complexity reduce the required overparameterization for optimal error rates.
• Achieved a fast O(1/n) risk rate under low-noise conditions for both network types.

Conclusions:

• The study provides a generalized understanding of GD generalization for deeper networks.
• Network scaling and complexity are key factors influencing generalization performance.
• The findings offer practical insights into training NNs for improved generalization and error rates.