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A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval

Zhong Zheng, Shiqian Ma, Lingzhou Xue

TL;DR

The paper tackles robust phase retrieval by formulating it as a nonsmooth, nonconvex optimization problem and introducing an inexact proximal linear (IPL) algorithm with two adaptive subproblem stopping criteria (LACC and HACC). The authors prove that IPL achieves $O(1/\epsilon^2)$ main iterations to obtain an $\epsilon$-stationary point under general conditions, and they establish local linear or quadratic convergence under a sharpness assumption. They implement the subproblem solver via an inexact FISTA (Nesterov) approach and provide thorough numerical results showing substantial speedups over the original PL method and subgradient approaches on synthetic and real data, including image-recovery tasks. The work highlights practical stopping criteria that preserve convergence while significantly reducing computation, with implications for robust nonconvex nonsmooth optimization beyond phase retrieval.

Abstract

This paper considers the robust phase retrieval problem, which can be cast as a nonsmooth and nonconvex optimization problem. We propose a new inexact proximal linear algorithm with the subproblem being solved inexactly. Our contributions are two adaptive stopping criteria for the subproblem. The convergence behavior of the proposed methods is analyzed. Through experiments on both synthetic and real datasets, we demonstrate that our methods are much more efficient than existing methods, such as the original proximal linear algorithm and the subgradient method.

A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval

TL;DR

The paper tackles robust phase retrieval by formulating it as a nonsmooth, nonconvex optimization problem and introducing an inexact proximal linear (IPL) algorithm with two adaptive subproblem stopping criteria (LACC and HACC). The authors prove that IPL achieves main iterations to obtain an -stationary point under general conditions, and they establish local linear or quadratic convergence under a sharpness assumption. They implement the subproblem solver via an inexact FISTA (Nesterov) approach and provide thorough numerical results showing substantial speedups over the original PL method and subgradient approaches on synthetic and real data, including image-recovery tasks. The work highlights practical stopping criteria that preserve convergence while significantly reducing computation, with implications for robust nonconvex nonsmooth optimization beyond phase retrieval.

Abstract

This paper considers the robust phase retrieval problem, which can be cast as a nonsmooth and nonconvex optimization problem. We propose a new inexact proximal linear algorithm with the subproblem being solved inexactly. Our contributions are two adaptive stopping criteria for the subproblem. The convergence behavior of the proposed methods is analyzed. Through experiments on both synthetic and real datasets, we demonstrate that our methods are much more efficient than existing methods, such as the original proximal linear algorithm and the subgradient method.
Paper Structure (14 sections, 18 theorems, 101 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 18 theorems, 101 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. IPL and Its Convergence Analysis
  3. Convergence under General Settings
  4. Local Convergence under Sharpness Assumption
  5. Related Work
  6. FISTA for Solving the Subproblem Inexactly
  7. Numerical Experiments
  8. Synthetic Data
  9. Image Recovery
  10. Proofs
  11. Proof of Theorem \ref{['thm:gen_convergence_rate']}
  12. Proof of Theorem \ref{['thm:convergence_rate_sharpness']}
  13. Proof of Theorem \ref{['overall_iplnesterov']}
  14. Conclusion

Key Result

Theorem 1

Denote $F^\star = \inf_{x\in\mathbb{R}^n} F(x)$. For Algorithm alg:adaptive-IPL with $t = 1/L$, the following conclusion holds.

Figures (3)

  • Figure 1: The comparison of success rates and CPU time on synthetic datasets with $p_{\textrm{fail}} = 0.05$ and $n=500$.
  • Figure 2: The comparison of success rates and CPU time on synthetic datasets with $p_{\textrm{fail}} = 0.15$ and $n=500$.
  • Figure 3: A real RNA nanoparticles image.

Theorems & Definitions (36)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 1: Weak Convexity
  • Lemma 2: see, e.g., equation (5.2) in drusvyatskiy2019efficiency
  • Lemma 3
  • proof : Proof of Lemma \ref{['thm:gen_descent_inexact0']}
  • ...and 26 more