mlsanityData to 7 Oct 2026

Subgradient methods for sharp weakly convex functions

arXiv 1803.02461 ↗First posted 6 Mar 2018math.OCcited by 33

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Abstract (arXiv)

Subgradient methods converge linearly on a convex function that grows sharply away from its solution set. In this work, we show that the same is true for sharp functions that are only weakly convex, provided that the subgradient methods are initialized within a fixed tube around the solution set. A variety of statistical and signal processing tasks come equipped with good initialization, and provably lead to formulations that are both weakly convex and sharp. Therefore, in such settings, subgradient methods can serve as inexpensive local search procedures. We illustrate the proposed techniques on phase retrieval and covariance estimation problems.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). First posted on arXiv 6 Mar 2018.

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Citing papers per year: 2021: 5; 2022: 2; 2023: 8; 2024: 9; 2025: 6; 2026: 3.

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  8. 2024Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisationcited by 12
  9. 2023Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimizationcited by 11
  10. 2025New Perspectives on the Polyak Stepsize: Surrogate Functions and Negative Resultscited by 6
  11. 2023Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penaltiescited by 4
  12. 2023Provably Convergent Data-Driven Convex-Nonconvex Regularizationcited by 4
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  3. 2024Disentangled Representation Learning with the Gromov-Monge Gapcited by 3
  4. 2021Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergencecited by 2
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  6. 2023A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functionscited by 2
  7. 2025Anchored Langevin Algorithmscited by 2
  8. 2025Safeguarded Stochastic Polyak Step Sizes for Non-smooth Optimization: Robust Performance Without Small (Sub)Gradientscited by 2
  9. 2021Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programmingcited by 1
  10. 2023Decentralized Weakly Convex Optimization Over the Stiefel Manifoldcited by 1
  11. 2023A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrievalcited by 1
  12. 2026Convergence of Spectral Descent for Non-smooth Optimizationcited by 1
  13. 2026Adaptive Sharpness-Aware Minimization with a Polyak-type Step size: A Theory-Grounded Schedulercited by 1
  14. 2021Complexity of Inexact Proximal Point Algorithm for minimizing convex functions with Holderian Growthcited by 0
  15. 2023An inexact LPA for DC composite optimization and application to matrix completions with outlierscited by 0
  16. 2024Multi-task learning via robust regularized clustering with non-convex group penaltiescited by 0
  17. 2024Smoothed Robust Phase Retrievalcited by 0
  18. 2024ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary pointscited by 0
  19. 2025Smoothing ADMM for Non-convex and Non-smooth Hierarchical Federated Learningcited by 0
  20. 2025Simplifying Optimal Transport through Schatten-p Regularizationcited by 0
  21. 2026Global Optimality for Constrained Exploration via Penalty Regularizationcited by 0