Median-Truncated Gradient Descent: A Robust and Scalable Nonconvex Approach for Signal Estimation

  • Yuejie ChiEmail author
  • Yuanxin Li
  • Huishuai Zhang
  • Yingbin Liang
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Recent work has demonstrated the effectiveness of gradient descent for directly estimating high-dimensional signals via nonconvex optimization in a globally convergent manner using a proper initialization. However, the performance is highly sensitive in the presence of adversarial outliers that may take arbitrary values. In this chapter, we introduce the median-Truncated Gradient Descent (median-TGD) algorithm to improve the robustness of gradient descent against outliers, and apply it to two celebrated problems: low-rank matrix recovery and phase retrieval. Median-TGD truncates the contributions of samples that deviate significantly from the sample median in each iteration in order to stabilize the search direction. Encouragingly, when initialized in a neighborhood of the ground truth known as the basin of attraction, median-TGD converges to the ground truth at a linear rate under Gaussian designs with a near-optimal number of measurements, even when a constant fraction of the measurements are arbitrarily corrupted. In addition, we introduce a new median-truncated spectral method that ensures an initialization in the basin of attraction. The stability against additional dense bounded noise is also established. Numerical experiments are provided to validate the superior performance of median-TGD.



The work of Y. Chi and Y. Li 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. The work of Y. Liang is supported in part by NSF under the grants CCF-1761506 and ECCS-1818904.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yuejie Chi
    • 1
    Email author
  • Yuanxin Li
    • 1
  • Huishuai Zhang
    • 2
  • Yingbin Liang
    • 3
  1. 1.Department of Electrical and Computer EngineeringCarnegie Mellon UniversityPittsburghUSA
  2. 2.Microsoft Research AsiaBeijingChina
  3. 3.Department of Electrical and Computer EngineeringThe Ohio State UniversityColumbusUSA

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