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Distributed Generalized Nash Equilibria Learning for Online Stochastic Aggregative Games

Kaixin Du, Min Meng

TL;DR

This work tackles online stochastic aggregative games with local decision-set constraints and time-varying coupled inequalities. It introduces a distributed online stochastic primal-dual push-sum algorithm that enables partial-information players to learn a variational GNE while tracking the aggregative variable over unbalanced, time-varying networks. The authors prove high-probability sublinear bounds on per-player regret R_i(T) and constraint violation R_g(T), and establish almost-sure convergence to the vGNE in time-invariant, strongly monotone cases, with sublinear rates for time-averaged iterates. The method is validated via simulations in an electricity-market-like scenario, demonstrating consensus, feasibility progress, and superior performance over baselines, indicating practical impact for real-time, distributed decision-making under uncertainty.

Abstract

This paper investigates online stochastic aggregative games subject to local set constraints and time-varying coupled inequality constraints, where each player possesses a time-varying expectation-valued cost function relying on not only its own decision variable but also an aggregation of all the players' variables. Each player can only access its local individual cost function and constraints, necessitating partial information exchanges with neighboring players through time-varying unbalanced networks. Additionally, local cost functions and constraint functions are not prior knowledge and only revealed gradually. To learn generalized Nash equilibria of such games, a novel distributed online stochastic algorithm is devised based on push-sum and primal-dual strategies. Through rigorous analysis, high probability bounds on the regret and constraint violation are provided by appropriately selecting decreasing stepsizes. Moreover, for a time-invariant stochastic strongly monotone game, it is shown that the generated sequence by the designed algorithm converges to its variational generalized Nash equilibrium (GNE) almost surely, and the time-averaged sequence converges sublinearly with high probability. Finally, the derived theoretical results are illustrated by numerical simulations.

Distributed Generalized Nash Equilibria Learning for Online Stochastic Aggregative Games

TL;DR

This work tackles online stochastic aggregative games with local decision-set constraints and time-varying coupled inequalities. It introduces a distributed online stochastic primal-dual push-sum algorithm that enables partial-information players to learn a variational GNE while tracking the aggregative variable over unbalanced, time-varying networks. The authors prove high-probability sublinear bounds on per-player regret R_i(T) and constraint violation R_g(T), and establish almost-sure convergence to the vGNE in time-invariant, strongly monotone cases, with sublinear rates for time-averaged iterates. The method is validated via simulations in an electricity-market-like scenario, demonstrating consensus, feasibility progress, and superior performance over baselines, indicating practical impact for real-time, distributed decision-making under uncertainty.

Abstract

This paper investigates online stochastic aggregative games subject to local set constraints and time-varying coupled inequality constraints, where each player possesses a time-varying expectation-valued cost function relying on not only its own decision variable but also an aggregation of all the players' variables. Each player can only access its local individual cost function and constraints, necessitating partial information exchanges with neighboring players through time-varying unbalanced networks. Additionally, local cost functions and constraint functions are not prior knowledge and only revealed gradually. To learn generalized Nash equilibria of such games, a novel distributed online stochastic algorithm is devised based on push-sum and primal-dual strategies. Through rigorous analysis, high probability bounds on the regret and constraint violation are provided by appropriately selecting decreasing stepsizes. Moreover, for a time-invariant stochastic strongly monotone game, it is shown that the generated sequence by the designed algorithm converges to its variational generalized Nash equilibrium (GNE) almost surely, and the time-averaged sequence converges sublinearly with high probability. Finally, the derived theoretical results are illustrated by numerical simulations.
Paper Structure (19 sections, 11 theorems, 92 equations, 9 figures, 1 algorithm)

This paper contains 19 sections, 11 theorems, 92 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Graph Theory
  4. Problem Formulation
  5. High Probability Bound
  6. MAIN RESULTS
  7. Numerical Simulation
  8. CONCLUSION
  9. Proof of Lemma \ref{['lem:mu-w-z']}
  10. Proof of Lemma \ref{['lem-consensus']}
  11. Proof of Lemma \ref{['lem:hp']}
  12. Proof of Lemma \ref{['lem:optimal']}
  13. Proof of Lemma \ref{['lem:mu-rel']}
  14. Proof of Lemma \ref{['lem-regret']}
  15. Proof of Theorem \ref{['thm:reg']}
  16. ...and 4 more sections

Key Result

Lemma 1

Under Assumptions ass:graph and ass:set, for any $t\geq 0$ and $i\in[N]$, there hold where $\underline{w}$ is given in Assumption ass:graph-i).

Figures (9)

  • Figure 1: Schematic illustration of four switching graphs.
  • Figure 2: Trajectories of $\mathcal{R}_i(T)/T$, $i\in[N]$ by Algorithm \ref{['alg:primaldual']}.
  • Figure 3: The trajectory of $\mathcal{R}_g(T)/T$ by Algorithm \ref{['alg:primaldual']}.
  • Figure 4: The trajectory of $g_t(x_t)$ by Algorithm \ref{['alg:primaldual']}.
  • Figure 5: Trajectories of consensus errors by Algorithm \ref{['alg:primaldual']}.
  • ...and 4 more figures

Theorems & Definitions (40)

  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 3
  • ...and 30 more