# Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer

arXiv: https://arxiv.org/abs/2109.02455 · HTML: https://mlsanity.com/paper/2109.02455/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 6 Sep 2021.

## Builds on (most cited first, 11 of 11)

- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems](https://mlsanity.com/paper/2006.11144/) (2020)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Convergence of constant step stochastic gradient descent for non-smooth non-convex functions](https://mlsanity.com/paper/2005.08513/) (2020)
- [Analysis of nonsmooth stochastic approximation: the differential inclusion approach](https://mlsanity.com/paper/1805.01916/) (2018)
- [Asymptotic study of stochastic adaptive algorithm in non-convex landscape](https://mlsanity.com/paper/2012.05640/) (2020)
- [Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance](https://mlsanity.com/paper/2012.04002/) (2020)
- [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)
- [A H"olderian backtracking method for min-max and min-min problems](https://mlsanity.com/paper/2007.08810/) (2020)
- [Stochastic proximal subgradient descent oscillates in the vicinity of its accumulation set](https://mlsanity.com/paper/2103.16253/) (2021)

## Built on by (most cited first, 1 of 1)

- [Nonconvex Penalized LAD Estimation in Partial Linear Models with DNNs: Asymptotic Analysis and Proximal Algorithms](https://mlsanity.com/paper/2511.21115/) (2025)

## Abstract (arXiv)

It was previously shown by Davis and Drusvyatskiy that every Clarke critical point of a generic, semialgebraic (and more generally definable in an o-minimal structure), weakly convex function is lying on an active manifold and is either a local minimum or an active strict saddle. In the first part of this work, we show that when the weak convexity assumption fails a third type of point appears: a sharply repulsive critical point. Moreover, we show that the corresponding active manifolds satisfy the Verdier and the angle conditions which were introduced by us in our previous work. In the second part of this work, we show that, under a density-like assumption on the perturbation sequence, the stochastic subgradient descent (SGD) avoids sharply repulsive critical points with probability one. We show that such a density-like assumption could be obtained upon adding a small random perturbation (e.g. a nondegenerate Gaussian) at each iteration of the algorithm. These results, combined with our previous work on the avoidance of active strict saddles, show that the SGD on a generic definable (e.g. semialgebraic) function converges to a local minimum.