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

arXiv: https://arxiv.org/abs/2108.11832 · HTML: https://mlsanity.com/paper/2108.11832/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 85 references that are in the map.
- First posted on arXiv 26 Aug 2021.

## Builds on (most cited first, 18 of 18)

- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems](https://mlsanity.com/paper/2002.07919/) (2020)
- [When Are Nonconvex Problems Not Scary?](https://mlsanity.com/paper/1510.06096/) (2015)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [The nonsmooth landscape of phase retrieval](https://mlsanity.com/paper/1711.03247/) (2017)
- [Stochastic algorithms with geometric step decay converge linearly on sharp functions](https://mlsanity.com/paper/1907.09547/) (2019)
- [The proximal point method revisited](https://mlsanity.com/paper/1712.06038/) (2017)
- [Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions](https://mlsanity.com/paper/2108.02072/) (2021)
- [Generic minimizing behavior in semi-algebraic optimization](https://mlsanity.com/paper/1504.07694/) (2015)
- [Orthogonal Invariance and Identifiability](https://mlsanity.com/paper/1304.1198/) (2013)
- [Variational analysis of spectral functions simplified](https://mlsanity.com/paper/1506.05170/) (2015)
- [Proximal methods avoid active strict saddles of weakly convex functions](https://mlsanity.com/paper/1912.07146/) (2019)
- [On a semismooth* Newton method for solving generalized equations](https://mlsanity.com/paper/1904.09167/) (2019)

## Built on by (most cited first, 11 of 11)

- [Asymptotic normality and optimality in nonsmooth stochastic approximation](https://mlsanity.com/paper/2301.06632/) (2023)
- [Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case](https://mlsanity.com/paper/2307.11782/) (2023)
- [Online Covariance Estimation in Nonsmooth Stochastic Approximation](https://mlsanity.com/paper/2502.05305/) (2025)
- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)
- [Piecewise Polynomial Regression of Tame Functions via Integer Programming](https://mlsanity.com/paper/2311.13544/) (2023)
- [Avoiding strict saddle points of nonconvex regularized problems](https://mlsanity.com/paper/2401.09274/) (2024)
- [On the Hardness of Meaningful Local Guarantees in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2409.10323/) (2024)
- [ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary points](https://mlsanity.com/paper/2410.05025/) (2024)
- [Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence](https://mlsanity.com/paper/2604.17067/) (2026)
- [Local linear convergence of gradient methods for overparameterized Gaussian mixtures](https://mlsanity.com/paper/2605.30936/) (2026)
- [Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Framework](https://mlsanity.com/paper/2608.03001/) (2026)

## 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.