mlsanityData to 7 Oct 2026

Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization

arXiv 2108.11832 ↗First posted 26 Aug 2021math.OC cs.LGcited by 11

Not on the map. Why?

Abstract (arXiv)

We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we present is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on a certain distinguished submanifold that captures the nonsmooth activity of the function. In the process, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier and extend stochastic processes techniques of Pemantle.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 18 of its 85 references that are in the map. First posted on arXiv 26 Aug 2021.

Builds on

18 of 85 references.

References · 18

  1. 2018Stochastic model-based minimization of weakly convex functionscited by 157
  2. 2017No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysiscited by 127
  3. 2016Matrix Completion has No Spurious Local Minimumcited by 124
  4. 2018Stochastic subgradient method converges on tame functionscited by 120
  5. 2016A Geometric Analysis of Phase Retrievalcited by 92
  6. 2016Global Optimality of Local Search for Low Rank Matrix Recoverycited by 81
  7. 2020Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problemscited by 39
  8. 2015When Are Nonconvex Problems Not Scary?cited by 27
  9. 2017Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrievalcited by 24
  10. 2017The nonsmooth landscape of phase retrievalcited by 16
  11. 2019Stochastic algorithms with geometric step decay converge linearly on sharp functionscited by 14
  12. 2017The proximal point method revisitedcited by 9
6 more
  1. 2021Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functionscited by 9
  2. 2015Generic minimizing behavior in semi-algebraic optimizationcited by 5
  3. 2013Orthogonal Invariance and Identifiabilitycited by 3
  4. 2015Variational analysis of spectral functions simplifiedcited by 3
  5. 2019Proximal methods avoid active strict saddles of weakly convex functionscited by 2
  6. 2019On a semismooth* Newton method for solving generalized equationscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 3; 2025: 2; 2026: 3.

11 papers.

  1. 2023Asymptotic normality and optimality in nonsmooth stochastic approximationcited by 14
  2. 2023Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Casecited by 13
  3. 2025Online Covariance Estimation in Nonsmooth Stochastic Approximationcited by 4
  4. 2025Deep Learning as the Disciplined Construction of Tame Objectscited by 2
  5. 2023Piecewise Polynomial Regression of Tame Functions via Integer Programmingcited by 1
  6. 2024Avoiding strict saddle points of nonconvex regularized problemscited by 1
  7. 2024On the Hardness of Meaningful Local Guarantees in Nonsmooth Nonconvex Optimizationcited by 0
  8. 2024ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary pointscited by 0
  9. 2026Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergencecited by 0
  10. 2026Local linear convergence of gradient methods for overparameterized Gaussian mixturescited by 0
  11. 2026Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Frameworkcited by 0