# Scaling Law Analysis in Federated Learning: How to Select the Optimal Model Size?

arXiv: https://arxiv.org/abs/2511.12188 · HTML: https://mlsanity.com/paper/2511.12188/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 48 references that are in the map.
- First posted on arXiv 15 Nov 2025.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Measuring the Effects of Non-Identical Data Distribution for Federated Visual Classification](https://mlsanity.com/paper/1909.06335/) (2019)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Language models scale reliably with over-training and on downstream tasks](https://mlsanity.com/paper/2403.08540/) (2024)
- [Scale Efficiently: Insights from Pre-training and Fine-tuning Transformers](https://mlsanity.com/paper/2109.10686/) (2021)
- [Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints](https://mlsanity.com/paper/1707.05947/) (2017)
- [Generalization Error Bounds for Noisy, Iterative Algorithms](https://mlsanity.com/paper/1801.04295/) (2018)
- [Personalized Federated Learning with Gaussian Processes](https://mlsanity.com/paper/2106.15482/) (2021)
- [Scaling Federated Learning for Fine-tuning of Large Language Models](https://mlsanity.com/paper/2102.00875/) (2021)
- [A PAC-Bayesian Analysis of Randomized Learning with Application to Stochastic Gradient Descent](https://mlsanity.com/paper/1709.06617/) (2017)
- [Scaling Language Model Size in Cross-Device Federated Learning](https://mlsanity.com/paper/2204.09715/) (2022)
- [Will LLMs Scaling Hit the Wall? Breaking Barriers via Distributed Resources on Massive Edge Devices](https://mlsanity.com/paper/2503.08223/) (2025)
- [Lessons from Generalization Error Analysis of Federated Learning: You May Communicate Less Often!](https://mlsanity.com/paper/2306.05862/) (2023)
- [Personalized Federated Learning of Probabilistic Models: A PAC-Bayesian Approach](https://mlsanity.com/paper/2401.08351/) (2024)

## Abstract (arXiv)

The recent success of large language models (LLMs) has sparked a growing interest in training large-scale models. As the model size continues to scale, concerns are growing about the depletion of high-quality, well-curated training data. This has led practitioners to explore training approaches like Federated Learning (FL), which can leverage the abundant data on edge devices while maintaining privacy. However, the decentralization of training datasets in FL introduces challenges to scaling large models, a topic that remains under-explored. This paper fills this gap and provides qualitative insights on generalizing the previous model scaling experience to federated learning scenarios. Specifically, we derive a PAC-Bayes (Probably Approximately Correct Bayesian) upper bound for the generalization error of models trained with stochastic algorithms in federated settings and quantify the impact of distributed training data on the optimal model size by finding the analytic solution of model size that minimizes this bound. Our theoretical results demonstrate that the optimal model size has a negative power law relationship with the number of clients if the total training compute is unchanged. Besides, we also find that switching to FL with the same training compute will inevitably reduce the upper bound of generalization performance that the model can achieve through training, and that estimating the optimal model size in federated scenarios should depend on the average training compute across clients. Furthermore, we also empirically validate the correctness of our results with extensive training runs on different models, network settings, and datasets.