Gradient Descent with Identity Initialization Efficiently Learns Positive-Definite Linear Transformations by Deep...

Area of Science:

• Machine Learning
• Deep Learning Theory
• Optimization Algorithms

Background:

• Deep linear neural networks offer a tractable model for understanding deep learning.
• Gradient descent is a fundamental optimization algorithm used in training neural networks.
• Analyzing convergence properties is crucial for developing reliable machine learning models.

Purpose of the Study:

• To analyze the convergence of gradient descent for function approximation using deep linear neural networks.
• To identify conditions under which gradient descent succeeds or fails in learning target matrices.
• To investigate the impact of initialization and regularization on learning performance.

Main Methods:

• Focus on gradient descent on population quadratic loss with isotropic input distributions.
• Derive polynomial iteration bounds for approximating the least-squares matrix.
• Examine scenarios with bounded excess loss and conditions for non-convergence.
• Analyze specific algorithms with regularization for symmetric and non-symmetric matrices.

Main Results:

• Polynomial convergence bounds are established when initial loss is sufficiently small.
• Gradient descent fails to converge when the target matrix is distant from identity or has negative eigenvalues.
• Certain regularization techniques do not guarantee convergence in problematic cases.
• A novel algorithm with specific regularizers shows polynomial convergence for non-symmetric matrices.

Conclusions:

• The success of gradient descent in deep linear networks is highly sensitive to the properties of the target matrix and initialization.
• Understanding these limitations is key to designing more robust deep learning algorithms.
• Further research into effective regularization and novel optimization strategies is warranted.