# A closed-measure approach to stochastic approximation

arXiv: https://arxiv.org/abs/2112.05482 · HTML: https://mlsanity.com/paper/2112.05482/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2021.

## Built on by (most cited first, 4 of 4)

- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)
- [Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2307.10053/) (2023)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions](https://mlsanity.com/paper/2403.11565/) (2024)

## Abstract (arXiv)

This paper introduces a new method to tackle the issue of the almost sure convergence of stochastic approximation algorithms defined from a differential inclusion. Under the assumption of slowly decaying step-sizes, we establish that the set of essential accumulation points of the iterates belongs to the Birkhoff center associated with the differential inclusion. Unlike previous works, our results do not rely on the notion of asymptotic pseudotrajectories introduced by Bena"im--Hofbauer--Sorin, which is the predominant technique to address the convergence problem. They follow as a consequence of Young's superposition principle for closed measures. This perspective bridges the gap between Young's principle and the notion of invariant measure of set-valued dynamical systems introduced by Faure and Roth. Also, the proposed method allows to obtain sufficient conditions under which the velocities locally compensate around any essential accumulation point.