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A Potential Game Perspective in Federated Learning

Kang Liu, Ziqi Wang, Enrique Zuazua

TL;DR

This study establishes the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings, and demonstrates a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server.

Abstract

Federated learning (FL) is an emerging paradigm for training machine learning models across distributed clients. Traditionally, in FL settings, a central server assigns training efforts (or strategies) to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To explore this, we propose a potential game framework where each client's payoff is determined by their individual efforts and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated through a reward factor. Our study begins by establishing the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings. We demonstrate a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server. Furthermore, we prove the convergence of the best-response algorithm to compute NEs for our FL game. Finally, we apply the training efforts derived from specific NEs to a real-world FL scenario, validating the effectiveness of the identified optimal reward factor.

A Potential Game Perspective in Federated Learning

TL;DR

This study establishes the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings, and demonstrates a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server.

Abstract

Federated learning (FL) is an emerging paradigm for training machine learning models across distributed clients. Traditionally, in FL settings, a central server assigns training efforts (or strategies) to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To explore this, we propose a potential game framework where each client's payoff is determined by their individual efforts and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated through a reward factor. Our study begins by establishing the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings. We demonstrate a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server. Furthermore, we prove the convergence of the best-response algorithm to compute NEs for our FL game. Finally, we apply the training efforts derived from specific NEs to a real-world FL scenario, validating the effectiveness of the identified optimal reward factor.

Paper Structure

This paper contains 21 sections, 8 theorems, 69 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. Related work
  4. Notations
  5. Outline of the paper.
  6. A potential game framework for federated learning
  7. Training-cost-incentive workflow in federated learning
  8. Existence of Nash equilibrium of $\Gamma_{\text{FL}}$
  9. Uniqueness results and the best-response algorithm
  10. Homogeneous game formulation
  11. Uniqueness of the NE of the homogeneous game
  12. Computation of NEs via the best-response algorithm
  13. Discussion on non-uniform data sizes and heterogeneous games
  14. Numerical experiments
  15. Setups
  16. ...and 6 more sections

Key Result

Theorem 2.4

The game $\Gamma_{\textnormal{FL}}$ is a (weighted) potential game with the potential given by and $w_i = 1/\rho_i$ for $i \in [m]$. If $\mathcal{S}$ is compact, then $\Gamma_{\textnormal{FL}}$ possesses at least one NE.

Figures (5)

  • Figure 1.1: Illustration of three critical reward factors: the activation point $\lambda_1$, the jump point $\lambda^*$, and the saturation point $\lambda_2$. See \ref{['cor_uniq']} for more details.
  • Figure 2.1: Training-cost-incentive workflow of FL, with the round index $t$ omitted.
  • Figure 4.1: Results of the clients' average training effort $\bar{s}$ with respect to the server's reward factor $\lambda$. Scenario 1 (Left): $q_i\in\{1\}$, $Q_i \in [20,30]$, $c_1 < c_2$. Scenario 2 (Right): $q_i\in[1,20]$, $Q_i \in [20,30]$, $c_1 > c_2$. The values of the activation point $\lambda_1$, jump point $\lambda^*$, and saturation point $\lambda_2$ are indicated in the legends.
  • Figure 4.2: Zoom-in of \ref{['fig_NE_zoom_out']} around the three critical thresholds: activation point $\lambda_1$, jump point $\lambda^*$, and saturation point $\lambda_2$. Scenario 1 (Top) corresponds to \ref{['fig_NE_zoom_out_a']}, and Scenario 2 (Bottom) corresponds to \ref{['fig_NE_zoom_out_b']}.
  • Figure 4.3: FL training performance in four cases with different server reward factors (see \ref{['tab_four_cases']}). Case 1 is selected before the activation point, Cases 2 and 3 are selected before and after the jump point, respectively, and Case 4 is selected after the saturation point.

Theorems & Definitions (33)

  • Definition 1.1: FL game $\Gamma_{\textnormal{FL}}$
  • Definition 1.2: Nash equilibrium
  • Definition 1.3: Potential game
  • Remark 2.1: Local training effort $s_i^t$ in \ref{['local_update']}
  • Remark 2.2: Quadratic structure of $c_i$ in \ref{['eq_cost_i']}
  • Remark 2.3: Quadratic structure of $r_i$ in \ref{['eq_reward1']}
  • Theorem 2.4: Existence
  • proof
  • Example 2.5: Total budget case
  • Example 2.6: Homogeneous case
  • ...and 23 more