Mathematical Programming

, Volume 176, Issue 1–2, pp 5–37 | Cite as

Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval

  • Yuxin Chen
  • Yuejie Chi
  • Jianqing Fan
  • Cong MaEmail author
Full Length Paper Series B


This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest \(\varvec{x}^{\natural }\in {\mathbb {R}}^{n}\) from m quadratic equations/samples \(y_{i}=(\varvec{a}_{i}^{\top }\varvec{x}^{\natural })^{2}, 1\le i\le m\). This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning. We investigate the efficacy 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.

Mathematics Subject Classification




Y. Chen is supported in part by the AFOSR YIP award FA9550-19-1-0030, by the ARO grant W911NF-18-1-0303, by the ONR grant N00014-19-1-2120, and by the Princeton SEAS innovation award. Y. Chi is supported in part by AFOSR under the grant FA9550-15-1-0205, by ONR under the grant N00014-18-1-2142, by ARO under the grant W911NF-18-1-0303, and by NSF under the grants CAREER ECCS-1818571 and CCF-1806154. J. Fan is supported in part by NSF grants DMS-1662139 and DMS-1712591 and NIH grant 2R01-GM072611-13.

Supplementary material

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Supplementary material 1 (pdf 746 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Department of Electrical and Computer EngineeringCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA

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