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Aggregative games with bilevel structures: Distributed algorithms and convergence analysis

Kaihong Lu, Huanshui Zhang, Long Wang

TL;DR

This work tackles distributed computation of Nash equilibria in aggregative games with bilevel structures, where the outer-level costs depend on an aggregation σ(x) defined by an inner-level minimization. It introduces two distributed schemes: a second-order gradient-based method (SOGD) that tracks Hessian information via a distributed average-tracking mechanism, and a first-order gradient-based method (FOGD) that estimates the necessary gradient product without Hessians using a two-point scheme. Under mild connectivity and regularity assumptions, both approaches guarantee convergence to the NE (or controlled approximation in the FO case) and reveal the associated convergence rates, with SOGD achieving $O(\sqrt{\ln t / t})$ and FO-convergence having additive bias terms scaling with the estimation parameters. Simulations on a distributed small-cell power allocation problem validate the theoretical results and illustrate effective aggregation-tracking and NE convergence in networked settings.

Abstract

In this paper, the problem of distributively seeking the equilibria of aggregative games with bilevel structures is studied. Different from the traditional aggregative games, here the aggregation is determined by the minimizer of a virtual leader's objective function in the inner level, which depends on the actions of the players in the outer level. Moreover, the global objective function of the virtual leader is formed by the sum of some local functions with two arguments, each of which is strongly convex with respect to the second argument. When making decisions, each player in the outer level only has access to a local part of the virtual leader's objective function. To handle this problem, first, we propose a second order gradient-based distributed algorithm, where the Hessian matrices associated with the objective functions of the leader are involved. By the algorithm, players update their actions while cooperatively minimizing the objective function of the virtual leader to estimate the aggregation by communicating with their neighbors via a connected graph. Under mild assumptions on the graph and cost functions, we prove that the actions of players asymptotically converge to the Nash equilibrium point. Then, for the case where the Hessian matrices associated with the objective functions of the virtual leader are not available, we propose a first order gradient-based distributed algorithm, where a distributed two-point estimate strategy is developed to estimate the gradients of players' cost functions in the outer level. Under the same conditions, we prove that the convergence errors of players' actions to the Nash equilibrium point are linear with respect to the estimate parameters. Finally, simulations are provided to demonstrate the effectiveness of our theoretical results.

Aggregative games with bilevel structures: Distributed algorithms and convergence analysis

TL;DR

This work tackles distributed computation of Nash equilibria in aggregative games with bilevel structures, where the outer-level costs depend on an aggregation σ(x) defined by an inner-level minimization. It introduces two distributed schemes: a second-order gradient-based method (SOGD) that tracks Hessian information via a distributed average-tracking mechanism, and a first-order gradient-based method (FOGD) that estimates the necessary gradient product without Hessians using a two-point scheme. Under mild connectivity and regularity assumptions, both approaches guarantee convergence to the NE (or controlled approximation in the FO case) and reveal the associated convergence rates, with SOGD achieving and FO-convergence having additive bias terms scaling with the estimation parameters. Simulations on a distributed small-cell power allocation problem validate the theoretical results and illustrate effective aggregation-tracking and NE convergence in networked settings.

Abstract

In this paper, the problem of distributively seeking the equilibria of aggregative games with bilevel structures is studied. Different from the traditional aggregative games, here the aggregation is determined by the minimizer of a virtual leader's objective function in the inner level, which depends on the actions of the players in the outer level. Moreover, the global objective function of the virtual leader is formed by the sum of some local functions with two arguments, each of which is strongly convex with respect to the second argument. When making decisions, each player in the outer level only has access to a local part of the virtual leader's objective function. To handle this problem, first, we propose a second order gradient-based distributed algorithm, where the Hessian matrices associated with the objective functions of the leader are involved. By the algorithm, players update their actions while cooperatively minimizing the objective function of the virtual leader to estimate the aggregation by communicating with their neighbors via a connected graph. Under mild assumptions on the graph and cost functions, we prove that the actions of players asymptotically converge to the Nash equilibrium point. Then, for the case where the Hessian matrices associated with the objective functions of the virtual leader are not available, we propose a first order gradient-based distributed algorithm, where a distributed two-point estimate strategy is developed to estimate the gradients of players' cost functions in the outer level. Under the same conditions, we prove that the convergence errors of players' actions to the Nash equilibrium point are linear with respect to the estimate parameters. Finally, simulations are provided to demonstrate the effectiveness of our theoretical results.

Paper Structure

This paper contains 23 sections, 14 theorems, 102 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Our contributions
  4. Notations
  5. Problem formulation
  6. Distributed methods involving second order gradient information
  7. Algorithm design
  8. Convergence results
  9. Distributed methods involving first order gradient information
  10. Algorithm design
  11. Convergence results
  12. A numerical simulation example
  13. Conclusions
  14. Appendix
  15. Proof of Proposition 1
  16. ...and 8 more sections

Key Result

Lemma 1

For game $\Gamma(\mathcal{V}, \Omega, {J})$, if $\Omega$ is convex and compact, and $J_i(x_i,$$\sigma(x))$, $i\in\mathcal{V}$ is differentiable and convex with respect to $x_i$, then the NEs always exist. Moreover, $x^*$ is an NE if and only if for any $k>0$,

Figures (6)

  • Figure 1: The AGs with bilevel structures.
  • Figure 2: The communication graph.
  • Figure 3: The trajectory of $E(t)$ under Algorithm 1.
  • Figure 4: The trajectories of $y_i(t)-y^*,~i$$=1,~\cdots,~20$ under Algorithm 1.
  • Figure 5: The trajectory of $E(t)$ under Algorithm 2.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Remark 1
  • Definition 1
  • Remark 2
  • Lemma 1: ChenFacchinei
  • Proposition 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Remark 3
  • Remark 4
  • ...and 23 more