# Stochastic Modified Equations for Continuous Limit of Stochastic ADMM

arXiv: https://arxiv.org/abs/2003.03532 · HTML: https://mlsanity.com/paper/2003.03532/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2020.

## Built on by (most cited first, 2 of 2)

- [Unbiased and Sign Compression in Distributed Learning: Comparing Noise Resilience via SDEs](https://mlsanity.com/paper/2502.17009/) (2025)
- [Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers](https://mlsanity.com/paper/2401.15838/) (2024)

## Abstract (arXiv)

Stochastic version of alternating direction method of multiplier (ADMM) and its variants (linearized ADMM, gradient-based ADMM) plays a key role for modern large scale machine learning problems. One example is the regularized empirical risk minimization problem. In this work, we put different variants of stochastic ADMM into a unified form, which includes standard, linearized and gradient-based ADMM with relaxation, and study their dynamics via a continuous-time model approach. We adapt the mathematical framework of stochastic modified equation (SME), and show that the dynamics of stochastic ADMM is approximated by a class of stochastic differential equations with small noise parameters in the sense of weak approximation. The continuous-time analysis would uncover important analytical insights into the behaviors of the discrete-time algorithm, which are non-trivial to gain otherwise. For example, we could characterize the fluctuation of the solution paths precisely, and decide optimal stopping time to minimize the variance of solution paths.