# Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization

arXiv: https://arxiv.org/abs/2407.02765 · HTML: https://mlsanity.com/paper/2407.02765/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 47 references that are in the map.
- First posted on arXiv 3 Jul 2024.

## Builds on (most cited first, 17 of 17)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [Distributed Stochastic Gradient Tracking Methods](https://mlsanity.com/paper/1805.11454/) (2018)
- [Mean Field Game of Controls and An Application To Trade Crowding](https://mlsanity.com/paper/1610.09904/) (2016)
- [Graphon Mean Field Games and the GMFG Equations](https://mlsanity.com/paper/2008.10216/) (2020)
- [Graphon Control of Large-scale Networks of Linear Systems](https://mlsanity.com/paper/1807.03412/) (2018)
- [The nonlinear heat equation on dense graphs and graph limits](https://mlsanity.com/paper/1302.5804/) (2013)
- [Linear-Quadratic Mean Field Games](https://mlsanity.com/paper/1404.5741/) (2014)
- [Linear-Quadratic N-person and Mean-Field Games with Ergodic Cost](https://mlsanity.com/paper/1401.1421/) (2014)
- [Graphon-based sensitivity analysis of SIS epidemics](https://mlsanity.com/paper/1912.10330/) (2019)
- [LQG Graphon Mean Field Games: Analysis via Graphon Invariant Subspaces](https://mlsanity.com/paper/2004.00679/) (2020)
- [Consensus Formation in First-Order Graphon Models with Time-Varying Topologies](https://mlsanity.com/paper/2111.03900/) (2021)
- [Propagation of Chaos of Forward-Backward Stochastic Differential Equations with Graphon Interactions](https://mlsanity.com/paper/2202.08163/) (2022)
- [Mean Field Games on Weighted and Directed Graphs via Colored Digraphons](https://mlsanity.com/paper/2209.03887/) (2022)
- [A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization](https://mlsanity.com/paper/1911.06665/) (2019)
- [Distributed Stochastic Optimization With Unbounded Subgradients Over Randomly Time-Varying Networks](https://mlsanity.com/paper/2008.08796/) (2020)
- [Graphon Particle Systems, Part I: Spatio-Temporal Approximation and Law of Large Numbers](https://mlsanity.com/paper/2405.16760/) (2024)

## Abstract (arXiv)

We study the distributed optimization problem over a graphon with a continuum of nodes, which is regarded as the limit of the distributed networked optimization as the number of nodes goes to infinity. Each node has a private local cost function. The global cost function, which all nodes cooperatively minimize, is the integral of the local cost functions on the node set. We propose stochastic gradient descent and gradient tracking algorithms over the graphon. We establish a general lemma for the upper bound estimation related to a class of time-varying differential inequalities with negative linear terms, based upon which, we prove that for both kinds of algorithms, the second moments of the nodes' states are uniformly bounded. Especially, for the stochastic gradient tracking algorithm, we transform the convergence analysis into the asymptotic property of coupled nonlinear differential inequalities with time-varying coefficients and develop a decoupling method. For both kinds of algorithms, we show that by choosing the time-varying algorithm gains properly, all nodes' states achieve L∞-consensus for a connected graphon. Furthermore, if the local cost functions are strongly convex, then all nodes' states converge to the minimizer of the global cost function and the auxiliary states in the stochastic gradient tracking algorithm converge to the gradient value of the global cost function at the minimizer uniformly in mean square.