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

Peter L Bartlett¹, David P Helmbold², Philip M Long³

¹Department of Statistics, University of California, Berkeley, Berkeley, CA 94720-3860, U.S.A. bartlett@cs.berkeley.edu.

Neural Computation

|January 16, 2019

Summary

This summary is machine-generated.

Gradient descent can approximate functions using deep linear neural networks, but convergence depends on the target matrix properties. Regularization may not always prevent failure, especially with negative eigenvalues.

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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.