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Multiplayer Federated Learning: Reaching Equilibrium with Less Communication

Conference: NeurIPS 2025 arXiv: 2501.08263 Code: None Area: Optimization Keywords: Federated Learning, Game Theory, Nash Equilibrium, Local SGD, Communication Efficiency

TL;DR

This paper proposes the Multiplayer Federated Learning (MpFL) framework, which models FL clients as rational players in a game-theoretic setting, and introduces the PEARL-SGD algorithm that reduces communication overhead via local updates while converging to a Nash equilibrium.

Background & Motivation

Root Cause

Background: Conventional federated learning assumes all clients share a collaborative objective and jointly optimize a single global model. In practice, however, participants in many scenarios are rational agents with individual objective functions that may conflict (e.g., Cournot competition, electricity markets, mobile robot control). Existing FL frameworks cannot handle such non-cooperative environments.

The authors identify three key observations: (1) classical FL's Local SGD cannot be directly applied to multiplayer games, since each player optimizes only its own objective; (2) federated minimax optimization involves a game but requires all clients to jointly adjust both sets of variables, rather than each client managing only its own; (3) personalized FL is a special case of MpFL, but MpFL is strictly more general.

Paper Goals

Overall Architecture

MpFL treats \(n\) clients as players in an \(n\)-player game.

Method

Overall Architecture

MpFL treats \(n\) clients as players in an \(n\)-player game. Each player \(i\) maintains its own strategy \(x^i \in \mathbb{R}^{d_i}\) and objective function \(f_i(x^1, \ldots, x^n)\), with the goal of finding a Nash equilibrium \(\mathbf{x}_\star\) such that no player can unilaterally improve its payoff. The equilibrium condition is \(f_i(x_\star^i; x_\star^{-i}) \leq f_i(x^i; x_\star^{-i}), \forall x^i, \forall i\). Communication is coordinated through a central server.

Key Designs

  1. PEARL-SGD Algorithm: Each player independently executes \(\tau\) steps of local SGD, fixing the strategies of all other players and updating only its own. A synchronization step is performed every \(\tau\) iterations — the server collects all players' strategies, concatenates them into a joint strategy vector, and broadcasts it back to all players. This differs from classical FL Local SGD in that each player computes gradients only with respect to its own variables.

  2. Player Drift Control: Analogous to client drift in classical FL, but fundamentally distinct. In MpFL, if the step size is not scaled with \(\tau\), local iterations of each player converge to a local minimum \(x_\star^i(x_{\tau p}^{-i})\) that depends on other players' strategies, potentially causing divergence. Consequently, the step size must satisfy the constraint \(\gamma = \mathcal{O}(1/\tau)\).

  3. Theoretical Assumption Framework: Convexity (CVX) and smoothness (SM) are assumed at the function level; quasi-strong monotonicity (QSM) and star cocoercivity (SCO) are assumed at the level of the joint gradient operator. QSM+SCO imply \(\mu \|\mathbf{x}-\mathbf{x}_\star\| \leq \|\mathcal{F}(\mathbf{x})\| \leq \ell \|\mathbf{x}-\mathbf{x}_\star\|\), with condition number \(\kappa = \ell/\mu\).

Loss & Training

Each player minimizes its own expected loss \(f_i(x^i; x^{-i}) = \mathbb{E}_{\xi^i \sim \mathcal{D}_i}[f_{i,\xi^i}(x^i; x^{-i})]\), using stochastic gradients \(g_k^i = \nabla f_{i,\xi_k^i}(x_k^i; x_{\tau p}^{-i})\). Both constant and decaying step size schedules are supported: constant step sizes yield convergence to a neighborhood of the equilibrium, while decaying step sizes guarantee exact convergence. Under constant step size, the neighborhood size is \(\mathcal{O}(\gamma \sigma^2 / \mu)\); decaying step size \(\gamma_k \propto 1/k\) achieves exact convergence at a sublinear rate.

Key Experimental Results

Main Results

Setup Metric Result Notes
Quadratic game (\(n=5\), \(d=10\)) \(\|\mathbf{x}_T - \mathbf{x}_\star\|^2\) Validates theoretical convergence rate Larger \(\tau\) reduces communication; convergence consistent with theory
Mobile robot control (\(n=4\)) Distance to equilibrium PEARL-SGD matches distributed GDA accuracy \(\tau=10\) reduces communication by 10× with comparable accuracy
Varying noise level \(\sigma\) Convergence neighborhood size \(\propto \sigma^2/\mu\) Fully consistent with Theorem 3.4
Varying player count \(n=2,5,10\) Convergence curves Stable convergence behavior Increasing \(n\) does not affect per-player convergence rate

Ablation Study

Configuration Metric Notes
\(\tau=1\) vs \(\tau=10\) vs \(\tau=50\) Convergence curves \(\tau=10\) achieves best communication–accuracy trade-off
Constant vs decaying step size Final distance Decaying step size enables exact convergence; constant step size converges to a neighborhood
Varying condition number \(\kappa\) Communication complexity Smaller \(\kappa\) allows larger \(\tau\) and greater communication savings

Key Findings

  • In the stochastic setting, PEARL-SGD reduces communication complexity from \(T\) to \(\Theta(\sqrt{T})\) under the condition \(\tau = \mathcal{O}(\sqrt{\mu T / L_{\max}})\).
  • In the deterministic setting, the step size constraint \(\gamma = \mathcal{O}(1/\tau)\) prevents PEARL-SGD from achieving communication savings.
  • Communication gains are more pronounced when \(L_{\max} \ll \ell\).

Highlights & Insights

  • This is the first work to introduce the Local SGD paradigm into the multiplayer game framework, establishing MpFL as a new research direction.
  • Player drift is a phenomenon unique to MpFL, fundamentally distinct from client drift in classical FL — it can cause divergence rather than merely yielding a suboptimal fixed point.
  • The theoretical analysis is tight: setting \(\tau=1\) recovers the standard SGDA analysis, consistent with existing literature.
  • The framework subsumes the consensus-regularization formulation of personalized FL as a special case.
  • Reducing communication complexity from \(T\) to \(\Theta(\sqrt{T})\) carries substantial practical significance for real distributed systems.
  • The role of the condition number \(\kappa = \ell/\mu\) is clear: smaller \(\kappa\) (better-conditioned problems) yields greater communication savings.
  • The heterogeneous data setting (with no distributional assumptions) enhances practical applicability.

Limitations & Future Work

  • The synchronization step incurs communication cost \(\mathcal{O}(D) = \mathcal{O}(d_1 + \cdots + d_n)\), which grows linearly with the number of players, making MpFL more suitable for cross-silo scenarios.
  • Gradient compression techniques to reduce per-round communication volume have not been integrated.
  • Mitigation strategies for player drift (e.g., variance correction analogous to SCAFFOLD) are left for future work.
  • Only the convex setting is considered; non-convex cases are not addressed.
  • Experiments are limited to synthetic data and simple control tasks, with no evaluation on practical ML benchmarks.
  • Partial participation (i.e., only a subset of players communicating per round) is not considered.
  • An asynchronous communication variant of PEARL-SGD remains unexplored.
  • The distinction from federated minimax optimization (Local SGDA/SEG) is that in the latter, all clients jointly optimize a single minimax problem, whereas in MpFL each client updates only its own variables.
  • The QSM and SCO assumptions are analogous to conditions in the variational inequality (VIP) literature and are weaker than strong monotonicity.
  • This work may inspire the application of FL techniques to distributed multi-agent systems and competitive market modeling.
  • The consensus-regularization formulation in personalized FL, \(\phi(x^1,\ldots,x^n) = \frac{\lambda}{2n}\sum_{i=1}^n \|x^i - \bar{x}\|^2\), is a subproblem of MpFL.
  • The concurrent work Decoupled SGD is algorithmically equivalent to PEARL-SGD but analyzes deterministic communication gains under a weakly coupled game assumption.
  • The distributed Nash equilibrium-seeking literature typically assumes cheap communication; MpFL is the first to introduce communication efficiency into this domain.
  • Future work may explore combining gradient compression (e.g., QSGD) with Local SGD to achieve dual communication savings.
  • MpFL in the non-convex non-monotone setting is an important open problem requiring new analytical tools.

Rating

  • Novelty: ⭐⭐⭐⭐ — Introduces a novel framework at the intersection of FL and game theory, with a well-motivated and practically meaningful problem formulation.
  • Experimental Thoroughness: ⭐⭐⭐ — Experiments are primarily for theoretical validation; large-scale real-world evaluations are lacking.
  • Writing Quality: ⭐⭐⭐⭐ — Notation is well-defined, theoretical derivations are rigorous, and tabular summaries are effective.
  • Value: ⭐⭐⭐⭐ — Opens a new direction in MpFL and establishes a theoretical foundation for communication-efficient distributed multiplayer game solving.