# On the Convergence of Distributed Stochastic Bilevel Optimization Algorithms over a Network

arXiv: https://arxiv.org/abs/2206.15025 · HTML: https://mlsanity.com/paper/2206.15025/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 45 references that are in the map.
- First posted on arXiv 30 Jun 2022.

## Builds on (most cited first, 28 of 28)

- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication](https://mlsanity.com/paper/1902.00340/) (2019)
- [On the Linear Speedup Analysis of Communication Efficient Momentum SGD for Distributed Non-Convex Optimization](https://mlsanity.com/paper/1905.03817/) (2019)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Distributed Stochastic Gradient Tracking Methods](https://mlsanity.com/paper/1805.11454/) (2018)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [PowerGossip: Practical Low-Rank Communication Compression in Decentralized Deep Learning](https://mlsanity.com/paper/2008.01425/) (2020)
- [Decentralized Gossip-Based Stochastic Bilevel Optimization over Communication Networks](https://mlsanity.com/paper/2206.10870/) (2022)
- [ZeroSARAH: Efficient Nonconvex Finite-Sum Optimization with Zero Full Gradient Computation](https://mlsanity.com/paper/2103.01447/) (2021)
- [Communication-Efficient Local Decentralized SGD Methods](https://mlsanity.com/paper/1910.09126/) (2019)
- [Decentralized Bilevel Optimization](https://mlsanity.com/paper/2206.05670/) (2022)
- [textttDeepSqueeze: Decentralization Meets Error-Compensated Compression](https://mlsanity.com/paper/1907.07346/) (2019)
- [Tighter Analysis of Alternating Stochastic Gradient Method for Stochastic Nested Problems](https://mlsanity.com/paper/2106.13781/) (2021)
- [A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization](https://mlsanity.com/paper/2102.06752/) (2021)
- [Periodic Stochastic Gradient Descent with Momentum for Decentralized Training](https://mlsanity.com/paper/2008.10435/) (2020)
- [Decentralized Submodular Maximization: Bridging Discrete and Continuous Settings](https://mlsanity.com/paper/1802.03825/) (2018)
- [Decentralized Frank-Wolfe Algorithm for Convex and Non-convex Problems](https://mlsanity.com/paper/1612.01216/) (2016)
- [Improving the Sample and Communication Complexity for Decentralized Non-Convex Optimization: A Joint Gradient Estimation and Tracking Approach](https://mlsanity.com/paper/1910.05857/) (2019)

## Built on by (most cited first, 14 of 14)

- [Decentralized Stochastic Bilevel Optimization with Improved per-Iteration Complexity](https://mlsanity.com/paper/2210.12839/) (2022)
- [Communication-Efficient Federated Bilevel Optimization with Local and Global Lower Level Problems](https://mlsanity.com/paper/2302.06701/) (2023)
- [A Single-Loop Algorithm for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2311.08945/) (2023)
- [Central Limit Theorem for Two-Timescale Stochastic Approximation with Markovian Noise: Theory and Applications](https://mlsanity.com/paper/2401.09339/) (2024)
- [A Penalty-Based Method for Communication-Efficient Decentralized Bilevel Programming](https://mlsanity.com/paper/2211.04088/) (2022)
- [Decentralized Multi-Level Compositional Optimization Algorithms with Level-Independent Convergence Rate](https://mlsanity.com/paper/2306.03322/) (2023)
- [Argus: Federated Non-convex Bilevel Learning over 6G Space-Air-Ground Integrated Network](https://mlsanity.com/paper/2505.09106/) (2025)
- [DICE: Data Influence Cascade in Decentralized Learning](https://mlsanity.com/paper/2507.06931/) (2025)
- [Nonconvex Decentralized Stochastic Bilevel Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2509.15543/) (2025)
- [DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity](https://mlsanity.com/paper/2606.21153/) (2026)
- [On the Communication Complexity of Decentralized Stochastic Bilevel Optimization](https://mlsanity.com/paper/2311.11342/) (2023)
- [A Communication and Computation Efficient Fully First-order Method for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2410.14115/) (2024)
- [S3LDBO: A Snapshot Single-Loop Algorithm for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2605.31311/) (2026)
- [Federated stochastic bilevel optimization with fully first-order gradients](https://mlsanity.com/paper/2609.16350/) (2026)

## Abstract (arXiv)

Bilevel optimization has been applied to a wide variety of machine learning models, and numerous stochastic bilevel optimization algorithms have been developed in recent years. However, most existing algorithms restrict their focus on the single-machine setting so that they are incapable of handling the distributed data. To address this issue, under the setting where all participants compose a network and perform peer-to-peer communication in this network, we developed two novel decentralized stochastic bilevel optimization algorithms based on the gradient tracking communication mechanism and two different gradient estimators. Additionally, we established their convergence rates for nonconvex-strongly-convex problems with novel theoretical analysis strategies. To our knowledge, this is the first work achieving these theoretical results. Finally, we applied our algorithms to practical machine learning models, and the experimental results confirmed the efficacy of our algorithms.