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Multiplayer Federated Learning: Reaching Equilibrium with Less Communication

TaeHo Yoon, Sayantan Choudhury, Nicolas Loizou

TL;DR

MpFL reinterprets federated learning as an n-player game where each client has its own objective, and introduces PEARL-SGD to update each player's action locally for τ steps before synchronizing via a central server. Under quasi-strong monotonicity and 1/ℓ-star-coercivity, PEARL-SGD achieves linear convergence to a Nash equilibrium in the deterministic regime and to a neighborhood in the stochastic regime, while reducing communication through infrequent synchronization. The paper provides detailed convergence guarantees, analyzes the trade-offs between synchronization interval τ, step-size γ, and data heterogeneity, and validates the theory with numerical experiments on a quadratic n-player game and distributed mobile robot control. The results demonstrate meaningful communication efficiency gains and illustrate the practical viability of MpFL for heterogeneous, strategic clients in federated settings.

Abstract

Traditional Federated Learning (FL) approaches assume collaborative clients with aligned objectives working towards a shared global model. However, in many real-world scenarios, clients act as rational players with individual objectives and strategic behaviors, a concept that existing FL frameworks are not equipped to adequately address. To bridge this gap, we introduce Multiplayer Federated Learning (MpFL), a novel framework that models the clients in the FL environment as players in a game-theoretic context, aiming to reach an equilibrium. In this scenario, each player tries to optimize their own utility function, which may not align with the collective goal. Within MpFL, we propose Per-Player Local Stochastic Gradient Descent (PEARL-SGD), an algorithm in which each player/client performs local updates independently and periodically communicates with other players. We theoretically analyze PEARL-SGD and prove that it reaches a neighborhood of equilibrium with less communication in the stochastic setup compared to its non-local counterpart. Finally, we verify our theoretical findings through numerical experiments.

Multiplayer Federated Learning: Reaching Equilibrium with Less Communication

TL;DR

MpFL reinterprets federated learning as an n-player game where each client has its own objective, and introduces PEARL-SGD to update each player's action locally for τ steps before synchronizing via a central server. Under quasi-strong monotonicity and 1/ℓ-star-coercivity, PEARL-SGD achieves linear convergence to a Nash equilibrium in the deterministic regime and to a neighborhood in the stochastic regime, while reducing communication through infrequent synchronization. The paper provides detailed convergence guarantees, analyzes the trade-offs between synchronization interval τ, step-size γ, and data heterogeneity, and validates the theory with numerical experiments on a quadratic n-player game and distributed mobile robot control. The results demonstrate meaningful communication efficiency gains and illustrate the practical viability of MpFL for heterogeneous, strategic clients in federated settings.

Abstract

Traditional Federated Learning (FL) approaches assume collaborative clients with aligned objectives working towards a shared global model. However, in many real-world scenarios, clients act as rational players with individual objectives and strategic behaviors, a concept that existing FL frameworks are not equipped to adequately address. To bridge this gap, we introduce Multiplayer Federated Learning (MpFL), a novel framework that models the clients in the FL environment as players in a game-theoretic context, aiming to reach an equilibrium. In this scenario, each player tries to optimize their own utility function, which may not align with the collective goal. Within MpFL, we propose Per-Player Local Stochastic Gradient Descent (PEARL-SGD), an algorithm in which each player/client performs local updates independently and periodically communicates with other players. We theoretically analyze PEARL-SGD and prove that it reaches a neighborhood of equilibrium with less communication in the stochastic setup compared to its non-local counterpart. Finally, we verify our theoretical findings through numerical experiments.
Paper Structure (60 sections, 14 theorems, 120 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 60 sections, 14 theorems, 120 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Multiplayer Federated Learning: Definition and Related Work
  4. Definition of MpFL
  5. Equilibrium in $n$-player game.
  6. MpFL.
  7. Assumptions on multiplayer game.
  8. Comparison with closely related FL frameworks
  9. Federated learning (FL).
  10. Federated minimax optimization.
  11. Personalized federated learning.
  12. PEARL-SGD: Algorithm and Convergence Guarantees
  13. Algorithm and assumptions
  14. Assumptions on the joint gradient operator.
  15. Convergence of PEARL-SGD: Deterministic setup
  16. ...and 45 more sections

Key Result

theorem 1

Assume assumption:convexity, assumption:smoothness, assumption:quasi-strong-monotonicity and assumption:star-cocoercivity. Let $L_{\mathrm{max}} = \max\{L_1, \dots, L_n\}$, $\kappa = \ell/\mu$ and $0 < \gamma_k \equiv \gamma \le \frac{1}{\ell\tau + 2(\tau - 1) L_{\mathrm{max}} \sqrt{\kappa}}$. Then where $\zeta = 2 - \gamma \ell\tau - 2(\tau - 1)\gamma L_{\mathrm{max}} \sqrt{\kappa/3} > 0$ (by th

Figures (6)

  • Figure 1: Illustration of MpFL for heterogeneous functions $f_i$. The goal is for each player to reach the equilibrium $\vx_\star = (x_\star^1, \dots, x_\star^n)$ (see \ref{['eqn:equilibrium']}) with as little communication as possible.
  • Figure 2: Performance plots for PEARL-SGD. Figures \ref{['fig:nplayer_det_theoretical_gamma_tau']} (deterministic) and \ref{['fig:nplayer_stoch_theoretical_gamma_tau']} (stochastic) show the relative error $\frac{\sqnorm{\vx_k - \vx_\star}}{\sqnorm{\vx_0 - \vx_\star}}$ on the $n$-player game defined by \ref{['eqn:n_player_objective']} with different values of $\tau$, using theoretical step-sizes. (We provide additional experiments for this $n$-player game setup in Appendix \ref{['section:additional-experiments']}.) Figure \ref{['subfig:robot-accuracy']} shows the relative error in the (stochastic) mobile robot control setup \ref{['eqn:mobile-robot-objectives']} for distinct values of $\tau$.
  • Figure 3: Heatmap of relative errors in logarithmic scale.
  • Figure 4: Plots of objective values $f_1, f_2$ in \ref{['eqn:appendix-simple-counterexample']} from running (left) Local SGD on the joint variable $(u,v)$ and (right)PEARL-SGD.
  • Figure 5: Performance plots for PEARL-SGD on the $n$-player game \ref{['eqn:n_player_objective']} with different values of $\tau$. For each $\tau$, we use the empirically tuned step-size $\gamma \in \{10^{-1}, 10^{-2}, \dots, 10^{-6}\}$ for the best relative error $\frac{\|\vx_{\tau p} - \vx_\star\|^2}{\|\vx_0 - \vx_\star\|^2}$. Figure \ref{['fig:nplayer_det_tuned_gamma_tau']} shows the result from deterministic setup and \ref{['fig:nplayer_stoch_tuned_gamma_tau']} shows the stochastic setup.
  • ...and 1 more figures

Theorems & Definitions (24)

  • theorem 1
  • theorem 2
  • corollary 1
  • theorem 3
  • lemma 1
  • lemma 2
  • lemma 3
  • proof : Proof outline for \ref{['theorem:stochastic-local-GDA']}
  • lemma 4
  • proof
  • ...and 14 more