# Distributed Stochastic Gradient Descent with Staleness: A Stochastic Delay Differential Equation Based Framework

arXiv: https://arxiv.org/abs/2406.11159 · HTML: https://mlsanity.com/paper/2406.11159/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 73 references that are in the map.
- First posted on arXiv 17 Jun 2024.

## Builds on (most cited first, 30 of 32)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Asynchronous Federated Optimization](https://mlsanity.com/paper/1903.03934/) (2019)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [The Roadmap to 6G -- AI Empowered Wireless Networks](https://mlsanity.com/paper/1904.11686/) (2019)
- [Is Local SGD Better than Minibatch SGD?](https://mlsanity.com/paper/2002.07839/) (2020)
- [Minibatch vs Local SGD for Heterogeneous Distributed Learning](https://mlsanity.com/paper/2006.04735/) (2020)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [Revisiting Distributed Synchronous SGD](https://mlsanity.com/paper/1702.05800/) (2017)
- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [Asynchronous Stochastic Gradient Descent with Delay Compensation](https://mlsanity.com/paper/1609.08326/) (2016)
- [LAG: Lazily Aggregated Gradient for Communication-Efficient Distributed Learning](https://mlsanity.com/paper/1805.09965/) (2018)
- [Error Compensated Quantized SGD and its Applications to Large-scale Distributed Optimization](https://mlsanity.com/paper/1806.08054/) (2018)
- [On the Validity of Modeling SGD with Stochastic Differential Equations (SDEs)](https://mlsanity.com/paper/2102.12470/) (2021)
- [Slow and Stale Gradients Can Win the Race: Error-Runtime Trade-offs in Distributed SGD](https://mlsanity.com/paper/1803.01113/) (2018)
- [Asynchronous SGD Beats Minibatch SGD Under Arbitrary Delays](https://mlsanity.com/paper/2206.07638/) (2022)
- [A Variational Analysis of Stochastic Gradient Algorithms](https://mlsanity.com/paper/1602.02666/) (2016)
- [Staleness-aware Async-SGD for Distributed Deep Learning](https://mlsanity.com/paper/1511.05950/) (2015)
- [Trust Region-Guided Proximal Policy Optimization](https://mlsanity.com/paper/1901.10314/) (2019)
- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [On Graduated Optimization for Stochastic Non-Convex Problems](https://mlsanity.com/paper/1503.03712/) (2015)
- [Non-Gaussianity of Stochastic Gradient Noise](https://mlsanity.com/paper/1910.09626/) (2019)
- [ASAGA: Asynchronous Parallel SAGA](https://mlsanity.com/paper/1606.04809/) (2016)
- [Differential Privacy Meets Federated Learning under Communication Constraints](https://mlsanity.com/paper/2101.12240/) (2021)
- [Gap Aware Mitigation of Gradient Staleness](https://mlsanity.com/paper/1909.10802/) (2019)
- [Asynchronous Distributed ADMM for Large-Scale Optimization- Part I: Algorithm and Convergence Analysis](https://mlsanity.com/paper/1509.02597/) (2015)
- [Stochastic Gradient MCMC with Stale Gradients](https://mlsanity.com/paper/1610.06664/) (2016)
- [Inexact Successive Quadratic Approximation for Regularized Optimization](https://mlsanity.com/paper/1803.01298/) (2018)
- [Differential Equations for Modeling Asynchronous Algorithms](https://mlsanity.com/paper/1805.02991/) (2018)
- [Faster Asynchronous SGD](https://mlsanity.com/paper/1601.04033/) (2016)

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

- [IGT-OMD: Implicit Gradient Transport for Decision-Focused Learning under Delayed Feedback](https://mlsanity.com/paper/2605.12693/) (2026)

## Abstract (arXiv)

Distributed stochastic gradient descent (SGD) has attracted considerable recent attention due to its potential for scaling computational resources, reducing training time, and helping protect user privacy in machine learning. However, the staggers and limited bandwidth may induce random computational/communication delays, thereby severely hindering the learning process. Therefore, how to accelerate asynchronous SGD by efficiently scheduling multiple workers is an important issue. In this paper, a unified framework is presented to analyze and optimize the convergence of asynchronous SGD based on stochastic delay differential equations (SDDEs) and the Poisson approximation of aggregated gradient arrivals. In particular, we present the run time and staleness of distributed SGD without a memorylessness assumption on the computation times. Given the learning rate, we reveal the relevant SDDE's damping coefficient and its delay statistics, as functions of the number of activated clients, staleness threshold, the eigenvalues of the Hessian matrix of the objective function, and the overall computational/communication delay. The formulated SDDE allows us to present both the distributed SGD's convergence condition and speed by calculating its characteristic roots, thereby optimizing the scheduling policies for asynchronous/event-triggered SGD. It is interestingly shown that increasing the number of activated workers does not necessarily accelerate distributed SGD due to staleness. Moreover, a small degree of staleness does not necessarily slow down the convergence, while a large degree of staleness will result in the divergence of distributed SGD. Numerical results demonstrate the potential of our SDDE framework, even in complex learning tasks with non-convex objective functions.