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

Safwan Labbi, Paul Mangold, Eric Moulines

arXiv:2610.11555v1cs.LG

TL;DR

Personalized federated learning asks whether one method can learn each agent's solution across unknown heterogeneity while gaining linear speedup when agents are sufficiently similar. PF-LSA mixes local and averaged updates, and the paper establishes personalized convergence with speedup under mild heterogeneity. Its TD-learning application uses stated generative-model and feature-design assumptions.

  • Problem

    Personalized learning seeks a method that reaches each agent's solution across unknown heterogeneity while achieving linear speedup when agents are sufficiently similar.

    Source: §1 · pp. 1–2

  • Method

    PF-LSA mixes each agent's stochastic update with the averaged update across agents, without imposing an explicit structure on agent similarity.

    Source: §1 · p. 2

  • Results

    PF-LSA converges to each agent's optimal solution and achieves linear speedup under mild heterogeneity.

    Source: §1 · p. 2, §7 · p. 10

  • Takeaways & Limitations

    A minimal mixing strategy is reported to combine full personalization with collaboration benefits without restrictive assumptions about agent similarity.

    Source: §1 · p. 2

    The TD-learning results require the generative-model assumption TD1 and an assumption on the feature-design matrix.

    Source: §5 · p. 7

Abstract

from arXiv · show

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.

1 INTRODUCTION

LSA can require many samples, while shared federated models may perform poorly when agents are heterogeneous. PF-LSA instead mixes local and averaged updates to preserve agent-specific solutions while using collaboration.

1 INTRODUCTION

  • LSA solves linear equations from stochastic estimates, but noise can make it sample-intensive when data collection is costly or distributed.The framework spans stochastic optimization, control and telecommunications, and reinforcement learning.Source: §1 · p. 1
  • Shared-model federated methods such as FedAvg can degrade under agent heterogeneity because one model may not fit every agent.Federated collaboration aims to improve sample efficiency by letting agents benefit from one another's data.Source: §1 · p. 1
  • Personalized federated methods target agent-specific solutions, but commonly rely on prior assumptions about heterogeneity structure or inter-agent similarity.Recent approaches that promote collaboration among similar agents require similarity knowledge unavailable in practice.Source: §1 · pp. 1–2
  • PF-LSA mixes each agent's update with the average update across agents without imposing an explicit structure on agent similarity.The paper reports that this simple approach uses available data while preserving personalization, with a single hyperparameter controlling the balance between local updates and classical federated averaging (McMahan et al., 2017). Its analysis identifies rapid consensus-error decay, slower disagreement decay that is smaller in more homogeneous settings, and linear speedup under mild heterogeneity.Source: §1 · p. 2

2 RELATED WORK

Existing federated LSA methods generally target a shared or mixture solution, while personalized federated methods often rely on structural assumptions or extra information. PF-LSA is presented as a simpler route to each agent's own solution while retaining collaboration.

2 RELATED WORK

  • Federated LSA and reinforcement-learning methods can accelerate learning, but the surveyed approaches target a single global or mixture solution rather than fully personalized solutions.FedAvg-like methods may diverge or bias the solution under heterogeneity; other approaches use control variates to address client drift or study mixture-environment solutions.Source: §2 · p. 2
  • Personalized federated methods include multi-task, clustering, and regularization approaches, with assumptions such as latent structure or well-separated clusters.Regularization methods can still optimize a global objective and require tuning penalty hyperparameters.Source: §2 · p. 3
  • Some recent control-variate methods target local solutions but require pairwise data-distribution ratios, whereas PF-LSA aggregates updates without such assumptions.PF-LSA is contrasted with approaches that require knowledge unavailable in practice.Source: §2 · p. 3

3 BACKGROUND ON PERSONALIZED FEDERATED LINEAR STOCHASTIC APPROXIMATION

The setup considers multiple agents with distinct linear systems and stochastic observations. It introduces conditions and quantities for analyzing operator behavior, stochasticity, and heterogeneity.

3 BACKGROUND ON PERSONALIZED FEDERATED LINEAR STOCHASTIC APPROXIMATION

  • Personalized federated LSA considers N agents, each with an individual linear system to solve.The passage frames the objective as solving each system individually.Source: §3 · p. 3
  • The local operators can differ across agents, and the analysis assumes a contraction condition for each operator.The stated condition follows prior work on LSA (Mou et al., 2020).Source: §3 · p. 3
  • The setup uses agent-specific stochastic observations to estimate the linear-system coefficients and defines quantities to track noise and heterogeneity.The text says these quantities measure stochasticity and heterogeneity for the analysis.Source: §3 · p. 3

4 THE PF-LSA ALGORITHM

PF-LSA combines each agent’s local stochastic update with the average update across agents, while retaining each agent’s personalized solution for any positive mixing parameter. Its convergence analysis separates fast consensus from slower disagreement, yielding linear speedup when heterogeneity is low and personalized convergence when heterogeneity is high.

4 THE PF-LSA ALGORITHM

  • Each agent combines its local stochastic update with the averaged update across agents, with β controlling the relative weight of local learning and collaboration.The method’s update mixes all agents’ information rather than selecting collaborators.Source: §4 · p. 4

4.1 PF-LSA ’S LIMIT POINTS ARE THE SOLUTIONS OF INDIVIDUAL SYSTEMS

  • For any β > 0, if PF-LSA converges, its limit is each agent’s individual optimal solution; as β approaches zero, it does not reduce to FedAvg.This permits small β values, which help control update variance.Source: §4.1 · pp. 4–5

4.2 NON-ASYMPTOTIC CONVERGENCE RATES

  • The refined error decomposition separates consensus from disagreement: consensus contracts quickly, while the slower personalization clock can dominate when heterogeneity is large.For any heterogeneity level, the general bound controls disagreement; in low-heterogeneity regimes, its contribution can become negligible.Source: §4.2 · pp. 5–6
  • When local operators are sufficiently similar and initial disagreement is negligible, PF-LSA achieves personalization with linear speedup and sample complexity eO(1/(Nϵ)).This favorable regime arises without tuning the mixing parameter.Source: §4.2 · p. 6
  • Under high heterogeneity, the slow personalization component is no longer negligible, so collaboration need not improve overall convergence, while each agent still converges to its own solution.PF-LSA recovers the statistical complexity of individualized learning, eO(1/ϵ).Source: §4.2 · p. 7

5 APPLICATION TO PERSONALIZED FEDERATED TD LEARNING

The section applies PF-LSA to personalized TD learning, where each agent estimates its own policy’s value function using a linear feature approximation. It derives sample-complexity guarantees for low- and high-heterogeneity regimes.

Application of PF-LSA to TD learning

  • Under TD1 and TD2, TD(0) updates satisfy the PF-LSA assumptions A1 and A2.The section cites Mangold et al. (2024) for the proof of this claim.Source: §5 · pp. 7–8
  • PF-LSA is instantiated for personalized TD learning, with each agent approximating its own policy value function as a linear combination of features.The local target is defined through a projected Bellman equation.Source: §5 · pp. 7–8
  • In the low-heterogeneity regime, PF-LSA achieves a linear-speedup rate while converging to each agent’s personalized TD solution when initial disagreement is negligible relative to initial consensus error.The favorable heterogeneity threshold depends on the discount factor and feature conditioning: shorter effective horizons and better-conditioned features allow greater heterogeneity.Source: §5 · p. 8
  • In the high-heterogeneity regime, PF-LSA still estimates each agent’s local value function to any desired precision, with the same accuracy dependence as independent TD learning.The guarantee is stated under TD1 and TD2.Source: §5 · p. 8

6 NUMERICAL EXPERIMENTS

The experiments evaluate PF-LSA on personalized TD(0) across varying reward and transition-kernel heterogeneity. PF-LSA reaches low error faster than single-agent LSA and retains low final error where FedLSA becomes biased.

6 NUMERICAL EXPERIMENTS

  • PF-LSA remains stable and reaches low final error across the tested heterogeneity levels, while FedLSA plateaus at large error as reward heterogeneity rises.For reward heterogeneity ζrew ∈{0.5, 1}, FedLSA’s error remains several orders of magnitude above PF-LSA’s.Source: §6 · p. 9
  • PF-LSA reaches the low-error regime in fewer iterations than single-agent LSA across homogeneous, reward-heterogeneous, and transition-kernel-heterogeneous settings.The paper attributes the acceleration to mixing updates, which reduces the effective variance by using stochastic updates collected across agents.Source: §6 · p. 9
  • PF-LSA combines collaboration with personalization, whereas single-agent LSA learns slowly without other agents’ samples and FedLSA’s shared solution becomes biased under heterogeneity.The section reports that PF-LSA typically matches or improves on single-agent LSA’s final accuracy while requiring fewer iterations to reach the same error.Source: §6 · p. 9

7 CONCLUSION AND PERSPECTIVES

The conclusion presents PF-LSA as a simple update-mixing method that preserves agent-specific solutions while benefiting from collaboration, with linear speedup in controlled-heterogeneity regimes. The appendix supports the convergence analysis through a Lyapunov-function argument.

7 CONCLUSION AND PERSPECTIVES

  • PF-LSA mixes local and global updates to find agent-specific optima without assumptions on heterogeneity, and achieves linear speedup when heterogeneity is controlled.The analysis decomposes error into consensus and disagreement components; the authors identify Markovian-noise extensions, warm starts, and preconditioning as future directions.Source: §7 · p. 10

A CONVERGENCE OF A LYAPUNOV FUNCTION

  • The appendix’s Lyapunov-function argument concludes by taking total expectations to establish the stated bound.It also gives a lemma showing that the required condition on λ can be met under a step-size condition.Source: A CONVERGENCE OF A LYAPUNOV FUNCTION · p. 17

B A TIGHTER ANALYSIS OF PF-LSA THROUGH CONSENSUS/DISAGREEMENT

The appendix derives a unified PF-LSA convergence bound and then specializes it to low- and high-heterogeneity regimes. It also gives regime-specific sample-complexity results.

B A TIGHTER ANALYSIS OF PF-LSA THROUGH CONSENSUS/DISAGREEMENT

C APPLICATION TO PERSONALIZED FEDERATED TD-LEARNING

This section specializes the preceding results to personalized federated TD learning and states guarantees for high- and low-heterogeneity regimes under TD1 and TD2.

TD setup

Personalized federated TD guarantees

  • Corollaries C.1 and C.2 state personalized federated TD results for the high- and low-heterogeneity regimes, respectively, under TD1 and TD2.The supplied passages do not include the corollaries’ full bounds.Source: C APPLICATION TO PERSONALIZED… · p. 25

Sample and communication complexity

  • Corollaries C.3 and C.4 address sample complexity in the low- and high-heterogeneity regimes, with C.4 also addressing communication complexity.Both use β = 1/N; the passages additionally state regime-specific conditions and step-size requirements without giving the full expressions.Source: C APPLICATION TO PERSONALIZED… · p. 25

D TECHNICAL LEMMAS

The technical lemmas use a consensus subspace and orthogonal projection to express a squared-distance decomposition, alongside auxiliary proof steps and a stated bound under assumptions A1 and A2.

Lemma D.1

  • The proof snippets for Lemma D.1 specify distinct real x and y and reindex the first sum in a derivation.Source: D TECHNICAL LEMMAS · p. 26

Consensus subspace

  • The consensus subspace E consists of vectors formed by repeating one vector across agents, and P is described as its orthogonal projector.Source: D TECHNICAL LEMMAS · p. 26

Projection geometry

  • The projection argument uses Pythagoras’ identity to split squared distance into projected and residual components.The displayed identity relates ||Px − Py||² to ||x − y||² and the difference between the residuals x − Px and y − Py.Source: D TECHNICAL LEMMAS · p. 26

Lemma D.3

  • Lemma D.3 states a bound for any agent and x under A1 and A2, for η in the stated interval.The supplied passage does not include the bound’s expression.Source: D TECHNICAL LEMMAS · p. 26

Figure 2 experiment

  • The Figure 2 description reports that added agents accelerate convergence in low heterogeneity, while collaboration’s benefit diminishes as heterogeneity rises.The figure’s curves report average MSE across agents over five independent runs.Source: D TECHNICAL LEMMAS · p. 27

E ADDITIONAL EXPERIMENTS

The experiments report that PF-LSA’s convergence acceleration from additional agents is strongest at low heterogeneity, while personalization persists across heterogeneity levels.

Low heterogeneity

  • In homogeneous and mildly heterogeneous settings, increasing N from 2 to 8 to 16 accelerates convergence, with N = 16 reaching the low-error regime earliest.The authors relate the speedup in sufficiently similar local problems to reduced effective stochastic variance from averaging information across agents.Source: E ADDITIONAL EXPERIMENTS · p. 27

Moderate and strong heterogeneity

  • Under moderate heterogeneity, more agents still improve transient convergence, but under strong heterogeneity the curves are closer and acceleration is limited.Across both settings, PF-LSA converges to each agent’s personalized solution.Source: E ADDITIONAL EXPERIMENTS · p. 27

Additional experiments

  • Figure 2 shows average MSE across agents; PF-LSA retains personalized convergence across heterogeneity levels, while the acceleration from more agents diminishes as heterogeneity rises.The low-heterogeneity settings show substantial acceleration with more agents, whereas strong heterogeneity yields only limited acceleration.Source: E ADDITIONAL EXPERIMENTS · pp. 27–28

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