# Understanding Outer Optimizers in Local SGD: Learning Rates, Momentum, and Acceleration

arXiv: https://arxiv.org/abs/2509.10439 · HTML: https://mlsanity.com/paper/2509.10439/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 65 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 30 of 54)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Adaptive Federated Optimization](https://mlsanity.com/paper/2003.00295/) (2020)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Federated Learning with Differential Privacy: Algorithms and Performance Analysis](https://mlsanity.com/paper/1911.00222/) (2019)
- [A Unified Theory of Decentralized SGD with Changing Topology and Local Updates](https://mlsanity.com/paper/2003.10422/) (2020)
- [A Field Guide to Federated Optimization](https://mlsanity.com/paper/2107.06917/) (2021)
- [Tighter Theory for Local SGD on Identical and Heterogeneous Data](https://mlsanity.com/paper/1909.04746/) (2019)
- [Don't Use Large Mini-Batches, Use Local SGD](https://mlsanity.com/paper/1808.07217/) (2018)
- [The Road Less Scheduled](https://mlsanity.com/paper/2405.15682/) (2024)
- [Federated Learning with Compression: Unified Analysis and Sharp Guarantees](https://mlsanity.com/paper/2007.01154/) (2020)
- [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)
- [Lower Bounds and Optimal Algorithms for Personalized Federated Learning](https://mlsanity.com/paper/2010.02372/) (2020)
- [DiLoCo: Distributed Low-Communication Training of Language Models](https://mlsanity.com/paper/2311.08105/) (2023)
- [MegaScale: Scaling Large Language Model Training to More Than 10,000 GPUs](https://mlsanity.com/paper/2402.15627/) (2024)
- [Petals: Collaborative Inference and Fine-tuning of Large Models](https://mlsanity.com/paper/2209.01188/) (2022)
- [Federated Accelerated Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08950/) (2020)
- [DoG is SGD's Best Friend: A Parameter-Free Dynamic Step Size Schedule](https://mlsanity.com/paper/2302.12022/) (2023)
- [FedExP: Speeding Up Federated Averaging via Extrapolation](https://mlsanity.com/paper/2301.09604/) (2023)
- [Distributed Deep Learning in Open Collaborations](https://mlsanity.com/paper/2106.10207/) (2021)
- [Communication-Efficient Adaptive Federated Learning](https://mlsanity.com/paper/2205.02719/) (2022)
- [Convergence and Accuracy Trade-Offs in Federated Learning and Meta-Learning](https://mlsanity.com/paper/2103.05032/) (2021)
- [Semi-Cyclic Stochastic Gradient Descent](https://mlsanity.com/paper/1904.10120/) (2019)
- [Sharp Bounds for Federated Averaging (Local SGD) and Continuous Perspective](https://mlsanity.com/paper/2111.03741/) (2021)

## Built on by (most cited first, 9 of 9)

- [SNOO: Step-K Nesterov Outer Optimizer - The Surprising Effectiveness of Nesterov Momentum Applied to Pseudo-Gradients](https://mlsanity.com/paper/2510.15830/) (2025)
- [Smoothing DiLoCo with Primal Averaging for Faster Training of LLMs](https://mlsanity.com/paper/2512.17131/) (2025)
- [Factored Gossip DiLoCo: Reducing Blocking Communication in DiLoCo](https://mlsanity.com/paper/2606.22768/) (2026)
- [High dimensional theory of two-phase optimizers](https://mlsanity.com/paper/2603.26954/) (2026)
- [Outer-Momentum Restarting in High-Dimensional Two-Phase Optimization](https://mlsanity.com/paper/2605.28585/) (2026)
- [Local MixVR: Breaking the Communication-Sample Dependence in Distributed Learning](https://mlsanity.com/paper/2606.01128/) (2026)
- [FoMoE: Breaking the Full-Replica Barrier with a Federation of MoEs](https://mlsanity.com/paper/2606.19025/) (2026)
- [What's in a Smoothness Constant? Tighter Rates for Local SGD with Bounded Second-order Heterogeneity](https://mlsanity.com/paper/2607.14731/) (2026)
- [AutoLoCo: Communication Efficient Distributed LLM Training via Adaptive Synchronization](https://mlsanity.com/paper/2609.36662/) (2026)

## Abstract (arXiv)

Modern machine learning often requires training with large batch size, distributed data, and massively parallel compute hardware (like mobile and other edge devices or distributed data centers). Communication becomes a major bottleneck in such settings but methods like Local Stochastic Gradient Descent (Local SGD) show great promise in reducing this additional communication overhead. Local SGD consists of three parts: a local optimization process, an aggregation mechanism, and an outer optimizer that uses the aggregated updates from the nodes to produce a new model. While there exists an extensive literature on understanding the impact of hyperparameters in the local optimization process, the choice of outer optimizer and its hyperparameters is less clear. We study the role of the outer optimizer in Local SGD, and prove new convergence guarantees for the algorithm. In particular, we show that tuning the outer learning rate allows us to (a) trade off between optimization error and stochastic gradient noise variance, and (b) make up for ill-tuning of the inner learning rate. Our theory suggests that the outer learning rate should sometimes be set to values greater than 1. We extend our results to settings where we use momentum in the outer optimizer, and we show a similar role for the momentum-adjusted outer learning rate. We also study acceleration in the outer optimizer and show that it improves the convergence rate as a function of the number of communication rounds, improving upon the convergence rate of prior algorithms that apply acceleration locally. Finally, we also introduce a novel data-dependent analysis of Local SGD that yields further insights on outer learning rate tuning. We conduct comprehensive experiments with standard language models and various outer optimizers to validate our theory.