# Convergence of constant step stochastic gradient descent for non-smooth non-convex functions

arXiv: https://arxiv.org/abs/2005.08513 · HTML: https://mlsanity.com/paper/2005.08513/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2020.

## Built on by (most cited first, 27 of 27)

- [Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion](https://mlsanity.com/paper/2302.03775/) (2023)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [Adam-family Methods for Nonsmooth Optimization with Convergence Guarantees](https://mlsanity.com/paper/2305.03938/) (2023)
- [Global Convergence and Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/2110.01663/) (2021)
- [Constant Stepsize Q-learning: Distributional Convergence, Bias and Extrapolation](https://mlsanity.com/paper/2401.13884/) (2024)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Properties of Discrete Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.10352/) (2023)
- [Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions](https://mlsanity.com/paper/2108.02072/) (2021)
- [Early Directional Convergence in Deep Homogeneous Neural Networks for Small Initializations](https://mlsanity.com/paper/2403.08121/) (2024)
- [Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements](https://mlsanity.com/paper/2306.16502/) (2023)
- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)
- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [Survey Descent: A Multipoint Generalization of Gradient Descent for Nonsmooth Optimization](https://mlsanity.com/paper/2111.15645/) (2021)
- [PAC-Bayesian Learning of Optimization Algorithms](https://mlsanity.com/paper/2210.11113/) (2022)
- [Convergence of SGD for Training Neural Networks with Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.11714/) (2023)
- [Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2307.10053/) (2023)
- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [Understanding Nonlinear Implicit Bias via Region Counts in Input Space](https://mlsanity.com/paper/2505.11370/) (2025)
- [Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer](https://mlsanity.com/paper/2109.02455/) (2021)
- [Improved Overparametrization Bounds for Global Convergence of Stochastic Gradient Descent for Shallow Neural Networks](https://mlsanity.com/paper/2201.12052/) (2022)
- [Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions](https://mlsanity.com/paper/2403.11565/) (2024)
- [Inexact subgradient methods for semialgebraic functions](https://mlsanity.com/paper/2404.19517/) (2024)
- [Hamiltonian Monte Carlo on ReLU Neural Networks is Inefficient](https://mlsanity.com/paper/2410.22065/) (2024)
- [Numerical influence of ReLU'(0) on backpropagation](https://mlsanity.com/paper/2106.12915/) (2021)
- [Tame Riemannian Stochastic Approximation](https://mlsanity.com/paper/2302.00709/) (2023)
- [Quantitative Convergence Analysis of Projected Stochastic Gradient Descent for Non-Convex Losses via the Goldstein Subdifferential](https://mlsanity.com/paper/2510.02735/) (2025)

## Abstract (arXiv)

This paper studies the asymptotic behavior of the constant step Stochastic Gradient Descent for the minimization of an unknown function F , defined as the expectation of a non convex, non smooth, locally Lipschitz random function. As the gradient may not exist, it is replaced by a certain operator: a reasonable choice is to use an element of the Clarke subdifferential of the random function; an other choice is the output of the celebrated backpropagation algorithm, which is popular amongst practionners, and whose properties have recently been studied by Bolte and Pauwels [7]. Since the expectation of the chosen operator is not in general an element of the Clarke subdifferential BF of the mean function, it has been assumed in the literature that an oracle of BF is available. As a first result, it is shown in this paper that such an oracle is not needed for almost all initialization points of the algorithm. Next, in the small step size regime, it is shown that the interpolated trajectory of the algorithm converges in probability (in the compact convergence sense) towards the set of solutions of the differential inclusion. Finally, viewing the iterates as a Markov chain whose transition kernel is indexed by the step size, it is shown that the invariant distribution of the kernel converge weakly to the set of invariant distribution of this differential inclusion as the step size tends to zero. These results show that when the step size is small, with large probability, the iterates eventually lie in a neighborhood of the critical points of the mean function F .