Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval.

Area of Science:

• Optimization Algorithms
• Signal Processing
• Machine Learning

Background:

• Phase retrieval, recovering objects from quadratic equations, is crucial in various scientific fields.
• Existing methods often require complex initialization or sample handling for solving these nonconvex problems.

Purpose of the Study:

• To analyze the effectiveness of gradient descent (Wirtinger flow) for solving phase retrieval problems.
• To establish global convergence guarantees for vanilla gradient descent in this context.

Main Methods:

• Investigated gradient descent applied to the nonconvex least squares formulation of phase retrieval.
• Employed Gaussian designs and a leave-one-out analysis to understand algorithm behavior.
• Leveraged statistical models for analyzing optimization algorithms.

Main Results:

• Proved that randomly initialized gradient descent achieves ϵ-accuracy in O(log n + log(1/ϵ)) iterations.
• Demonstrated near-optimal computational and sample complexities simultaneously.
• Established the first global convergence guarantee for vanilla gradient descent in phase retrieval.

Conclusions:

• Gradient descent, with random initialization, offers an efficient and robust solution for phase retrieval.
• The findings eliminate the need for specialized initialization, sample splitting, or advanced saddle-point escaping techniques.
• This work provides a theoretical foundation for using standard optimization techniques in complex signal recovery problems.