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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$y_{i}=(\mathbf{a}_{i}^{\top}\mathbf{x}^{\natural})^{2}$, $1\leq i\leq m$. Thisproblem, also dubbed as phase retrieval, spans multiple domains includingphysical 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 Gaussiandesigns, gradient descent --- when randomly initialized --- yields an$\epsilon$-accurate solution in $O\big(\log n+\log(1/\epsilon)\big)$ iterationsgiven nearly minimal samples, thus achieving near-optimal computational andsample complexities at once. This provides the first global convergenceguarantee concerning vanilla gradient descent for phase retrieval, without theneed of (i) carefully-designed initialization, (ii) sample splitting, or (iii)sophisticated saddle-point escaping schemes. All of these are achieved byexploiting the statistical models in analyzing optimization algorithms, via aleave-one-out approach that enables the decoupling of certain statisticaldependency between the gradient descent iterates and the data.

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