Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval
This provides the first global convergence guarantee for vanilla gradient descent in phase retrieval, eliminating the need for specialized initialization or techniques, which is significant for applications in physical sciences and machine learning.
The paper tackles the nonconvex phase retrieval problem by proving that gradient descent with random initialization achieves near-optimal computational and sample complexities, specifically converging to an ε-accurate solution in O(log n + log(1/ε)) iterations under Gaussian designs with nearly minimal samples.
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 $ε$-accurate solution in $O\big(\log n+\log(1/ε)\big)$ 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.