# Minimax Estimation for Personalized Federated Learning: An Alternative between FedAvg and Local Training?

arXiv: https://arxiv.org/abs/2103.01901 · HTML: https://mlsanity.com/paper/2103.01901/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2021.

## Built on by (most cited first, 15 of 15)

- [Personalized Federated Learning with Gaussian Processes](https://mlsanity.com/paper/2106.15482/) (2021)
- [FedAvg with Fine Tuning: Local Updates Lead to Representation Learning](https://mlsanity.com/paper/2205.13692/) (2022)
- [Adaptive and Robust Multi-Task Learning](https://mlsanity.com/paper/2202.05250/) (2022)
- [Understanding Generalization of Federated Learning via Stability: Heterogeneity Matters](https://mlsanity.com/paper/2306.03824/) (2023)
- [FLIX: A Simple and Communication-Efficient Alternative to Local Methods in Federated Learning](https://mlsanity.com/paper/2111.11556/) (2021)
- [Federated Asymptotics: a model to compare federated learning algorithms](https://mlsanity.com/paper/2108.07313/) (2021)
- [FedGKD: Unleashing the Power of Collaboration in Federated Graph Neural Networks](https://mlsanity.com/paper/2309.09517/) (2023)
- [Minibatch and Local SGD: Algorithmic Stability and Linear Speedup in Generalization](https://mlsanity.com/paper/2310.01139/) (2023)
- [Personalized Federated Learning with Multiple Known Clusters](https://mlsanity.com/paper/2204.13619/) (2022)
- [Mitigating Privacy-Utility Trade-off in Decentralized Federated Learning via f-Differential Privacy](https://mlsanity.com/paper/2510.19934/) (2025)
- [Collaborative Learning with Different Labeling Functions](https://mlsanity.com/paper/2402.10445/) (2024)
- [Understanding the Statistical Accuracy-Communication Trade-off in Personalized Federated Learning with Minimax Guarantees](https://mlsanity.com/paper/2410.08934/) (2024)
- [Heterogeneity Matters even More in Distributed Learning: Study from Generalization Perspective](https://mlsanity.com/paper/2503.01598/) (2025)
- [On ADMM in Heterogeneous Federated Learning: Personalization, Robustness, and Fairness](https://mlsanity.com/paper/2407.16397/) (2024)
- [Learning Theory of the SVRG: Generalization and Convergence Analysis](https://mlsanity.com/paper/2605.28513/) (2026)

## Abstract (arXiv)

A widely recognized difficulty in federated learning arises from the statistical heterogeneity among clients: local datasets often originate from distinct yet not entirely unrelated probability distributions, and personalization is, therefore, necessary to achieve optimal results from each individual's perspective. In this paper, we show how the excess risks of personalized federated learning using a smooth, strongly convex loss depend on data heterogeneity from a minimax point of view, with a focus on the FedAvg algorithm (McMahan et al., 2017) and pure local training (i.e., clients solve empirical risk minimization problems on their local datasets without any communication). Our main result reveals an approximate alternative between these two baseline algorithms for federated learning: the former algorithm is minimax rate optimal over a collection of instances when data heterogeneity is small, whereas the latter is minimax rate optimal when data heterogeneity is large, and the threshold is sharp up to a constant. As an implication, our results show that from a worst-case point of view, a dichotomous strategy that makes a choice between the two baseline algorithms is rate-optimal. Another implication is that the popular FedAvg following by local fine tuning strategy is also minimax optimal under additional regularity conditions. Our analysis relies on a new notion of algorithmic stability that takes into account the nature of federated learning.