From Convergence to Generalization: Stability of Stationary-Point Learning Algorithms

Area of Science:

• Machine Learning
• Learning Theory
• Optimization

Background:

• Algorithmic stability is crucial for generalization guarantees in learning theory.
• Classical stability analyses often rely on convexity assumptions, limiting their applicability.
• Nonconvex settings present challenges for traditional stability analysis.

Purpose of the Study:

• To investigate algorithmic stability and generalization in nonconvex settings.
• To develop new theoretical bounds applicable to a wider range of learning algorithms.
• To provide a framework for understanding algorithm behavior beyond simple convergence.

Main Methods:

• Introduced an algorithm-dependent quantity based on the training dataset and algorithm output.
• Established stability and generalization bounds under mild differentiability assumptions.
• Applied the general framework to gradient descent for linear models and shallow neural networks.

Main Results:

• Developed bounds that explicitly incorporate optimization error and algorithm-dependent quantity.
• Bounds capture local curvature of the objective function.
• Analysis remains valid even if the algorithm does not converge to a minimizer.

Conclusions:

• The proposed framework offers robust stability and generalization bounds for nonconvex machine learning.
• The findings have implications for understanding gradient descent in various models.
• Empirical studies confirm the effectiveness of the developed stability analyses.