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Personalized Collaborative Learning with Affinity-Based Variance Reduction

Chenyu Zhang, Navid Azizan

TL;DR

This work tackles the challenge of learning personalized solutions in heterogeneous multi-agent systems while collaborating through a central server. It introduces Affinity-Based Personalized Collaborative Learning (AffPCL), which combines personalized bias correction with density-ratio-based importance correction to adaptively harness collaboration when agents are similar and revert to independent learning when they are not. The authors establish a finite-sample convergence rate of $O\left( t^{-1} \cdot \max\{ n^{-1}, \tilde\delta\}\right)$, where $\tilde\delta$ captures effective heterogeneity and stochastic conditioning, showing affinity-based variance reduction that interpolates between FL speedups and independent learning. The framework supports online central objective estimation and central objective learning, and demonstrates agent-specific speedups, including linear speedup for agents near a virtual center even under high heterogeneity. The results have practical implications for scalable, personalized decision making in federated and multi-agent settings, with future work pointing to privacy, communication efficiency, and broader affinity structures.

Abstract

Multi-agent learning faces a fundamental tension: leveraging distributed collaboration without sacrificing the personalization needed for diverse agents. This tension intensifies when aiming for full personalization while adapting to unknown heterogeneity levels -- gaining collaborative speedup when agents are similar, without performance degradation when they are different. Embracing the challenge, we propose personalized collaborative learning (PCL), a novel framework for heterogeneous agents to collaboratively learn personalized solutions with seamless adaptivity. Through carefully designed bias correction and importance correction mechanisms, our method AffPCL robustly handles both environment and objective heterogeneity. We prove that AffPCL reduces sample complexity over independent learning by a factor of $\max\{n^{-1}, δ\}$, where $n$ is the number of agents and $δ\in[0,1]$ measures their heterogeneity. This affinity-based acceleration automatically interpolates between the linear speedup of federated learning in homogeneous settings and the baseline of independent learning, without requiring prior knowledge of the system. Our analysis further reveals that an agent may obtain linear speedup even by collaborating with arbitrarily dissimilar agents, unveiling new insights into personalization and collaboration in the high heterogeneity regime.

Personalized Collaborative Learning with Affinity-Based Variance Reduction

TL;DR

This work tackles the challenge of learning personalized solutions in heterogeneous multi-agent systems while collaborating through a central server. It introduces Affinity-Based Personalized Collaborative Learning (AffPCL), which combines personalized bias correction with density-ratio-based importance correction to adaptively harness collaboration when agents are similar and revert to independent learning when they are not. The authors establish a finite-sample convergence rate of , where captures effective heterogeneity and stochastic conditioning, showing affinity-based variance reduction that interpolates between FL speedups and independent learning. The framework supports online central objective estimation and central objective learning, and demonstrates agent-specific speedups, including linear speedup for agents near a virtual center even under high heterogeneity. The results have practical implications for scalable, personalized decision making in federated and multi-agent settings, with future work pointing to privacy, communication efficiency, and broader affinity structures.

Abstract

Multi-agent learning faces a fundamental tension: leveraging distributed collaboration without sacrificing the personalization needed for diverse agents. This tension intensifies when aiming for full personalization while adapting to unknown heterogeneity levels -- gaining collaborative speedup when agents are similar, without performance degradation when they are different. Embracing the challenge, we propose personalized collaborative learning (PCL), a novel framework for heterogeneous agents to collaboratively learn personalized solutions with seamless adaptivity. Through carefully designed bias correction and importance correction mechanisms, our method AffPCL robustly handles both environment and objective heterogeneity. We prove that AffPCL reduces sample complexity over independent learning by a factor of , where is the number of agents and measures their heterogeneity. This affinity-based acceleration automatically interpolates between the linear speedup of federated learning in homogeneous settings and the baseline of independent learning, without requiring prior knowledge of the system. Our analysis further reveals that an agent may obtain linear speedup even by collaborating with arbitrarily dissimilar agents, unveiling new insights into personalization and collaboration in the high heterogeneity regime.
Paper Structure (43 sections, 25 theorems, 174 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 43 sections, 25 theorems, 174 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Personalization falls short in federated learning
  4. Slower rates in independent learning
  5. Curse of heterogeneity in collaborative learning
  6. Problem formulation
  7. Terminology and notation
  8. Roadmap
  9. Warm-up: Heterogeneous federated learning
  10. Constants
  11. Introducing personalization: Personalized bias correction
  12. Bias correction
  13. Control variates
  14. Central-local decomposition
  15. Introducing adaptivity: Central objective estimation
  16. ...and 28 more sections

Key Result

proposition 1

With a constant step size $\alpha \equiv \ln t /(\lambda t)$, eq:fl satisfies where $\widetilde{O}$ suppresses the logarithmic dependence on $\ln t$.The $\ln t$ dependence can be removed by using a linearly diminishing step size and considering a convex combination of the iterates $\{x^{c}_{\tau}\}_{\tau=0}^{t}$, as specified in lem:step. This refinement applies to all result

Figures (4)

  • Figure 1: matches in the homogeneous setting and independent learning in the high heterogeneity regime. Across all scenarios, consistently achieves the lowest MSE, while the other two methods' relative performance varies with the heterogeneity level. In the high heterogeneity regime where all agents are dissimilar, the agent closest to the center still enjoys significant speedup.
  • Figure 2: Improvement of .
  • Figure 3: Iso-performance contours of .
  • Figure 4: Agent-specific performance.

Theorems & Definitions (51)

  • proposition 1
  • definition 1: Objective heterogeneity
  • proposition 2
  • definition 2: Environment heterogeneity
  • theorem 1
  • theorem 2
  • definition 3: Stochastic condition number
  • proof
  • proof
  • proof
  • ...and 41 more