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Random Reshuffling-Based Distributed Nash Equilibrium Seeking

Jun Hu, Chao Sun, Chen Bo, Jianzheng Wang, Zheming Wang

Abstract

This paper studies random reshuffling (RR)-based distributed Nash equilibrium seeking for noncooperative games. The game is motivated as a sample-average approximation of an underlying expected-value stochastic game, while the algorithmic focus is placed on the resulting finite-sum equilibrium problem. Unlike existing distributed stochastic Nash equilibrium methods that mainly rely on with-replacement sampling, the proposed approach incorporates without-replacement component updates into equilibrium computation over networks. We first consider a full-information benchmark, for which an intermediate reference trajectory and a shuffling variance are introduced to characterize the epoch-wise dynamics induced by RR. The method is then extended to the more practical partial-decision-information setting, where each player updates its action using local estimates of the joint action profile. For the full-information case, a descent-type bound is established for the RR iterates. For the distributed partial-decision-information case, it is shown that, under constant parameters, the proposed algorithm converges linearly to a neighborhood of the Nash equilibrium, while under diminishing parameters, it converges exactly to the Nash equilibrium almost surely and in mean square. Numerical experiments on an EV charging game and a nonquadratic edge resource admission game demonstrate that RR consistently outperforms the conventional with-replacement SGD baseline in both steady-state accuracy and long-horizon performance.

Random Reshuffling-Based Distributed Nash Equilibrium Seeking

Abstract

This paper studies random reshuffling (RR)-based distributed Nash equilibrium seeking for noncooperative games. The game is motivated as a sample-average approximation of an underlying expected-value stochastic game, while the algorithmic focus is placed on the resulting finite-sum equilibrium problem. Unlike existing distributed stochastic Nash equilibrium methods that mainly rely on with-replacement sampling, the proposed approach incorporates without-replacement component updates into equilibrium computation over networks. We first consider a full-information benchmark, for which an intermediate reference trajectory and a shuffling variance are introduced to characterize the epoch-wise dynamics induced by RR. The method is then extended to the more practical partial-decision-information setting, where each player updates its action using local estimates of the joint action profile. For the full-information case, a descent-type bound is established for the RR iterates. For the distributed partial-decision-information case, it is shown that, under constant parameters, the proposed algorithm converges linearly to a neighborhood of the Nash equilibrium, while under diminishing parameters, it converges exactly to the Nash equilibrium almost surely and in mean square. Numerical experiments on an EV charging game and a nonquadratic edge resource admission game demonstrate that RR consistently outperforms the conventional with-replacement SGD baseline in both steady-state accuracy and long-horizon performance.

Paper Structure

This paper contains 15 sections, 7 theorems, 51 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. non-cooperative game and nash equilibrium
  4. random reshuffling
  5. Random Reshuffling Based Nash Equilibrium Seeking
  6. Simulation
  7. Numerical Experiments
  8. EV Charging Game
  9. Edge Resource Admission Game
  10. Conclusion
  11. Proof of Lemma 1
  12. Proof of Lemma 2
  13. Proof of Theorem 2
  14. Proof of Collary 1
  15. Proof of Theorem 3

Key Result

Lemma 1

By the smoothness and convexity of $f_i$ assumed in Assumption fi, $\sigma_{i,\text{shuffle}}^2$ can be bounded by where $\sigma^2_{\star}=\frac{1}{mn}\sum_{i=1}^n\sum_{\ell=1}^m\|\nabla f_i(\mathbf{x}_\star;\ell)\|^2$ $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Key Difference between RR and SGD
  • Figure 2: Convergence comparison for the EV charging game under constant and diminishing stepsizes.
  • Figure 3: Convergence comparison for the edge resource admission game under constant and diminishing stepsizes.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Corollary 1
  • ...and 2 more