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Finite-time convergence to an $ε$-efficient Nash equilibrium in potential games

Anna Maddux, Reda Ouhamma, Maryam Kamgarpour

TL;DR

This work delivers the first finite-time convergence guarantees for log-linear learning to an $\epsilon$-efficient Nash equilibrium in general potential games by establishing a mixing-time bound based on the log-Sobolev constant. The results show a polynomial dependence on $1/\epsilon$ (via a problem-dependent gap $\Delta$) and an exponential dependence on the number of players $N$, with a symmetry-based corollary yielding a polynomial-in-$N$ bound. It also analyzes practical variants, including binary feedback and perturbed/noisy settings, proving robustness and near-equivalence in convergence speed to the full-information case under mild conditions. The findings have practical implications for decentralized learning in multi-agent systems, where convergence speed to efficiently coordinated outcomes is critical. Overall, the paper advances the theoretical understanding and practical applicability of decentralized learning in potential games.

Abstract

This paper investigates the convergence time of log-linear learning to an $ε$-efficient Nash equilibrium in potential games, where an efficient Nash equilibrium is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. We prove the first finite-time convergence to an $ε$-efficient Nash equilibrium in general potential games. Our bounds depend polynomially on $1/ε$, an improvement over previous bounds for subclasses of potential games that are exponential in $1/ε$. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning requiring a constant factor less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.

Finite-time convergence to an $ε$-efficient Nash equilibrium in potential games

TL;DR

This work delivers the first finite-time convergence guarantees for log-linear learning to an -efficient Nash equilibrium in general potential games by establishing a mixing-time bound based on the log-Sobolev constant. The results show a polynomial dependence on (via a problem-dependent gap ) and an exponential dependence on the number of players , with a symmetry-based corollary yielding a polynomial-in- bound. It also analyzes practical variants, including binary feedback and perturbed/noisy settings, proving robustness and near-equivalence in convergence speed to the full-information case under mild conditions. The findings have practical implications for decentralized learning in multi-agent systems, where convergence speed to efficiently coordinated outcomes is critical. Overall, the paper advances the theoretical understanding and practical applicability of decentralized learning in potential games.

Abstract

This paper investigates the convergence time of log-linear learning to an -efficient Nash equilibrium in potential games, where an efficient Nash equilibrium is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. We prove the first finite-time convergence to an -efficient Nash equilibrium in general potential games. Our bounds depend polynomially on , an improvement over previous bounds for subclasses of potential games that are exponential in . We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning requiring a constant factor less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.
Paper Structure (34 sections, 11 theorems, 111 equations, 2 figures, 1 table)

This paper contains 34 sections, 11 theorems, 111 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Problem setup
  5. Convergence of log-linear learning
  6. Algorithm and background
  7. Convergence time in general potential games
  8. Convergence time in symmetric potential games
  9. Proof of Theorem \ref{['thm:convergence_rates']}
  10. Proof of Lemma \ref{['lem:general_bound_log_Sobolev']}:
  11. Comparison of $X_t^*$ and $X_t$:
  12. Comparison of $X_t$ and $\hat{X}_t$:
  13. Comparison of $\hat{X}_t$ and $\Bar{X}_t$:
  14. Combination:
  15. Robustness of log-linear learning
  16. ...and 19 more sections

Key Result

Theorem 3.1

Consider a potential game with a potential function $\Phi:\mathcal{A}^N\rightarrow [0,1]$ and with $A\ge 4$.We assume $A\ge 4$ to bound the log-Sobolev constant in Lemma lem:general_bound_log_Sobolev. For $\epsilon\in(0,1)$ and initial distribution $\mu^0$, assume that players adhere to log-linear l Then, for $t\geq \frac{25N^2 A^5}{16\pi^2 }e^{4\beta}(\log\log A^N+\log \beta+2\log\frac{4}{\epsil

Figures (2)

  • Figure 1: Expected potential value when all players follow log-linear learning with $\beta$ set as the lower-bound of Inequality \ref{['eq:beta']}. Lines are averages over 30 randomly generated games, shaded areas represent one standard deviation, and stars mark the first time the desired precision $1-\epsilon$ is reached. Top row is given for fixed precision $\epsilon = 0.05$ and various suboptimality gaps $\Delta$, and bottom row for fixed suboptimality gap $\Delta=0.1$ and various precisions $\epsilon$.
  • Figure 2: Comparison of log-linear learning. Lines are averages over 30 randomly generated games, shaded areas represent one standard deviation, and stars mark the first time the desired precision $1-\epsilon$ is reached. The results are given for a fixed precision $\epsilon=0.05$

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Definition 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 5.1
  • Theorem 5.2
  • ...and 4 more