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

TL;DR

This paper proves that simple gradient descent with random initialization can efficiently solve the nonconvex phase retrieval problem, achieving near-optimal sample and computational complexity without special initialization or saddle-point techniques.

Contribution

It provides the first global convergence guarantee for vanilla gradient descent in phase retrieval under Gaussian models, removing the need for careful initialization or advanced algorithms.

Findings

Gradient descent converges to an accurate solution in logarithmic iterations.

Achieves near-optimal sample complexity with minimal samples.

No specialized initialization or saddle-point escaping needed.

Abstract

This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest $\mathbf{x}^{\natural}\in\mathbb{R}^{n}$ from $m$ quadratic equations/samples $y_{i}=(\mathbf{a}_{i}^{\top}\mathbf{x}^{\natural})^{2}$, $1\leq i\leq m$. This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning. We investigate the efficiency of gradient descent (or Wirtinger flow) designed for the nonconvex least squares problem. We prove that under Gaussian designs, gradient descent --- when randomly initialized --- yields an $\epsilon$-accurate solution in $O\big(\log n+\log(1/\epsilon)\big)$ iterations given nearly minimal samples, thus achieving near-optimal computational and sample complexities at once. This provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes. All of these are achieved by exploiting the statistical models in analyzing optimization algorithms, via a leave-one-out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data.