[Paper Review] Distributed Forward-Backward algorithms for stochastic generalized Nash equilibrium seeking

Barbara Franci, Sergio Grammatico|arXiv (Cornell University)|Dec 9, 2019

Extremum Seeking Control Systems37 references4 citations

TL;DR

This paper proposes a distributed, preconditioned forward–backward algorithm for solving stochastic generalized Nash equilibrium problems (SGNEPs) with expected-value cost functions. By approximating the pseudogradient via stochastic samples and leveraging cocoercivity, the method achieves almost sure convergence to a generalized Nash equilibrium under mild monotonicity assumptions, outperforming existing methods in numerical speed.

ABSTRACT

We consider the stochastic generalized Nash equilibrium problem (SGNEP) with expected-value cost functions. Inspired by Yi and Pavel (Automatica, 2019), we propose a distributed GNE seeking algorithm based on the preconditioned forward-backward operator splitting for SGNEP, where, at each iteration, the expected value of the pseudogradient is approximated via a number of random samples. As main contribution, we show almost sure convergence of our proposed algorithm if the pseudogradient mapping is restricted (monotone and) cocoercive. For non-generalized SNEPs, we show almost sure convergence also if the pseudogradient mapping is restricted strictly monotone. Numerical simulations show that the proposed forward-backward algorithm seems faster that other available algorithms.

Motivation & Objective

• To address the challenge of computing generalized Nash equilibria in large-scale, uncertain multi-agent systems with shared constraints.
• To develop a distributed algorithm where each agent only uses local information and noisy samples of the pseudogradient.
• To establish almost sure convergence for SGNEPs under restricted cocoercivity and for SNEPs under restricted strict monotonicity.
• To improve computational efficiency over existing stochastic methods like extragradient or forward–backward–forward by using a simpler, sample-based forward–backward scheme.

Proposed method

• The algorithm uses a preconditioned forward–backward splitting framework to solve the variational inequality formulation of the SGNEP.
• At each iteration, the expected-value pseudogradient is approximated using a sample average of i.i.d. realizations of the random variable.
• A stochastic error term is introduced to model the sampling noise, and its second moment is bounded using concentration inequalities.
• The method employs a diminishing stepsize rule and ensures boundedness of iterates via a Lyapunov-type analysis.
• Convergence is established using a projected fixed-point iteration with a residual measure that vanishes almost surely.
• The analysis relies on nonexpansiveness of projections and Young’s inequality to derive recursive bounds on the distance to the equilibrium.

Experimental results

Research questions

• RQ1 Can a distributed, stochastic forward–backward algorithm achieve almost sure convergence to a generalized Nash equilibrium in SGNEPs with only local information and sampled pseudogradient estimates?
• RQ2 What assumptions on the pseudogradient mapping (e.g., cocoercivity, monotonicity) are sufficient to guarantee convergence in the stochastic setting?
• RQ3 How does the sampling error from the stochastic approximation affect the convergence rate and stability of the algorithm?
• RQ4 Can the proposed algorithm be made more computationally efficient than existing methods like extragradient or forward–backward–forward under the same assumptions?
• RQ5 Does the preconditioning strategy enable faster convergence compared to standard forward–backward schemes in stochastic GNEP settings?

Key findings

• The proposed algorithm achieves almost sure convergence to a generalized Nash equilibrium when the pseudogradient mapping is restricted cocoercive.
• For non-generalized SNEPs, almost sure convergence is also established under restricted strict monotonicity of the pseudogradient.
• The sampling error is shown to decay as $\mathcal{O}(1/\sqrt{S_k})$, where $S_k$ is the number of samples at iteration $k$, ensuring asymptotic consistency.
• Numerical simulations indicate that the algorithm converges faster than other available stochastic GNEP solvers, despite its simplicity.
• The boundedness of the iterates and the vanishing residual imply that any cluster point of the sequence is a solution to the SGNEP.
• The convergence proof relies on a novel application of the preconditioned forward–backward splitting with stochastic approximation, extending deterministic results to the stochastic setting.