[Paper Review] Tracking-based distributed equilibrium seeking for aggregative games

Guido Carnevale, Filippo Fabiani+3 more|arXiv (Cornell University)|Oct 26, 2022

Advanced Thermodynamics and Statistical Mechanics44 references4 citations

TL;DR

This paper proposes two fully distributed, discrete-time algorithms for generalized Nash equilibrium (GNE) seeking in aggregative games over networks, combining projected pseudo-gradient methods with tracking mechanisms to reconstruct aggregative and coupling constraint variables. The key contribution is linear convergence to the GNE under partial information, with the first such result for coupling constraints in distributed settings.

ABSTRACT

We propose fully-distributed algorithms for Nash equilibrium seeking in aggregative games over networks. We first consider the case where local constraints are present and we design an algorithm combining, for each agent, (i) the projected pseudo-gradient descent and (ii) a tracking mechanism to locally reconstruct the aggregative variable. To handle coupling constraints arising in generalized settings, we propose another distributed algorithm based on (i) a recently emerged augmented primal-dual scheme and (ii) two tracking mechanisms to reconstruct, for each agent, both the aggregative variable and the coupling constraint satisfaction. Leveraging tools from singular perturbations analysis, we prove linear convergence to the Nash equilibrium for both schemes. Finally, we run extensive numerical simulations to confirm the effectiveness of our methods and compare them with state-of-the-art distributed equilibrium-seeking algorithms.

Motivation & Objective

• To design fully distributed algorithms for generalized Nash equilibrium (GNE) seeking in aggregative games where agents only access local information and communicate locally.
• To address the challenge of reconstructing global variables—aggregative variable and coupling constraint status—when agents lack full network knowledge.
• To achieve linear convergence to the GNE under partial information, particularly in the presence of coupling constraints.
• To extend existing methods by introducing a novel tracking-based augmented primal-dual scheme that ensures exact convergence with constant step-size.

Proposed method

• Combines projected pseudo-gradient descent with a local tracking mechanism to estimate the aggregative variable when local constraints are present.
• Introduces two auxiliary tracking variables—one for the aggregative variable and one for the coupling constraint multiplier—enabling distributed reconstruction in generalized games.
• Employs an augmented primal-dual scheme inspired by continuous-time centralized optimization, adapted to discrete-time distributed dynamics.
• Uses singular perturbations analysis to establish convergence properties, leveraging Lyapunov-like functions with matrix-weighted norms.
• Applies a system-theoretic approach to prove linear convergence by constructing a Lyapunov function that satisfies contraction conditions.
• Employs a constant step-size in the algorithm, ensuring exact convergence without diminishing step-sizes, unlike prior works.

Experimental results

Research questions

• RQ1 Can a fully distributed algorithm achieve linear convergence to the generalized Nash equilibrium in aggregative games with coupling constraints?
• RQ2 How can agents reconstruct global aggregative and constraint variables using only local information and neighbor communication?
• RQ3 What is the role of tracking mechanisms in enabling distributed equilibrium computation without centralized coordination?
• RQ4 Can a constant step-size scheme ensure exact convergence in distributed GNE seeking, avoiding the slower convergence of diminishing step-sizes?
• RQ5 How does the proposed tracking-based augmented primal-dual scheme compare in convergence rate and robustness to existing distributed algorithms?

Key findings

• The proposed algorithm for games with coupling constraints achieves linear convergence to the generalized Nash equilibrium, a first in the literature for distributed schemes with constant step-size.
• Linear convergence is proven via singular perturbations analysis and a Lyapunov function with matrix-weighted norm, ensuring contraction under appropriate step-size selection.
• The algorithm guarantees exact convergence without requiring diminishing step-sizes, improving on prior methods that only achieve approximate or sublinear convergence.
• Numerical simulations confirm the effectiveness of the method and demonstrate superior convergence speed compared to state-of-the-art distributed equilibrium-seeking algorithms.
• The convergence rate is quantitatively bounded by a contraction factor dependent on system parameters, with explicit dependence on the step-size and problem condition numbers.
• The method is robust to network topology and maintains performance even under partial information and communication constraints.