# Best of Both Worlds in Federated LSA: Speedup When Possible, Personalization Always

arXiv: https://arxiv.org/abs/2610.11555 · HTML: https://mlsanity.com/paper/2610.11555/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 6 of 6)

- [Federated Learning with Non-IID Data](https://mlsanity.com/paper/1806.00582/) (2018)
- [Three Approaches for Personalization with Applications to Federated Learning](https://mlsanity.com/paper/2002.10619/) (2020)
- [Federated Temporal Difference Learning with Linear Function Approximation under Environmental Heterogeneity](https://mlsanity.com/paper/2302.02212/) (2023)
- [Adaptive collaboration for online personalized distributed learning with heterogeneous clients](https://mlsanity.com/paper/2507.06844/) (2025)
- [Convergence Guarantees for Federated SARSA with Local Training and Heterogeneous Agents](https://mlsanity.com/paper/2512.17688/) (2025)
- [Personalized Collaborative Learning with Affinity-Based Variance Reduction](https://mlsanity.com/paper/2510.16232/) (2025)

## Abstract (arXiv)

We study personalized federated linear stochastic approximation (LSA), a framework which notably encompass personalized temporal difference learning. In this setting, heterogeneous agents collaborate to solve distinct linear fixed-point equations, each corresponding to an agent-specific learning problem. A central open question in personalized learning is whether a single method can adapt to an unknown level of heterogeneity by converging to each agent's personalized solution in all regimes while achieving a linear speedup in the number of agents when their learning problems are sufficiently similar. We answer this question affirmatively by introducing PF-LSA, a minimalist algorithm that mixes each agent's local stochastic update with the average update across agents, at no additional computational cost relative to standard federated methods. We prove that PF-LSA, achieves best-of-both-worlds guarantees without any prior knowledge on the level of heterogeneity. Our analysis is based on a sharp decomposition of the error into consensus and disagreement components. The consensus error decays rapidly, whereas the disagreement error decays more slowly but becomes negligible in low-heterogeneity regimes.