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Inverse learning of black-box aggregator for robust Nash equilibrium

Guanpu Chen, Gehui Xu, Fengxiang He, Dacheng Tao, Thomas Parisini, Karl Henrik Johansson

TL;DR

This work tackles robust Nash equilibrium computation in multi-player aggregative games with uncertain coupling, where the aggregator weights are hidden behind a black box. It introduces an inverse variational inequality (VI) framework to learn the unknown weights $\boldsymbol\beta$ from data pairs $(\boldsymbol\alpha, \mathbf{x}^*_{\boldsymbol\alpha})$ produced by an existing solver, and then reformulates the problem into a deterministic robust counterpart to obtain an rGNE. The authors derive a first-order (KKT-type) condition for the rGNE and establish a generalization bound ensuring the learned weights generalize to unseen uncertainty, supported by numerical experiments on a demand-response-like setting. The approach enables robust, data-driven handling of uncertainty in aggregative games with black-box aggregators, providing practical procedures and theoretical guarantees for real-world applications.

Abstract

In this note, we investigate the robustness of Nash equilibria (NE) in multi-player aggregative games with coupling constraints. There are many algorithms for computing an NE of an aggregative game given a known aggregator. When the coupling parameters are affected by uncertainty, robust NE need to be computed. We consider a scenario where players' weight in the aggregator is unknown, making the aggregator kind of "a black box". We pursue a suitable learning approach to estimate the unknown aggregator by proposing an inverse variational inequality-based relationship. We then utilize the counterpart to reconstruct the game and obtain first-order conditions for robust NE in the worst case. Furthermore, we characterize the generalization property of the learning methodology via an upper bound on the violation probability. Simulation experiments show the effectiveness of the proposed inverse learning approach.

Inverse learning of black-box aggregator for robust Nash equilibrium

TL;DR

This work tackles robust Nash equilibrium computation in multi-player aggregative games with uncertain coupling, where the aggregator weights are hidden behind a black box. It introduces an inverse variational inequality (VI) framework to learn the unknown weights from data pairs produced by an existing solver, and then reformulates the problem into a deterministic robust counterpart to obtain an rGNE. The authors derive a first-order (KKT-type) condition for the rGNE and establish a generalization bound ensuring the learned weights generalize to unseen uncertainty, supported by numerical experiments on a demand-response-like setting. The approach enables robust, data-driven handling of uncertainty in aggregative games with black-box aggregators, providing practical procedures and theoretical guarantees for real-world applications.

Abstract

In this note, we investigate the robustness of Nash equilibria (NE) in multi-player aggregative games with coupling constraints. There are many algorithms for computing an NE of an aggregative game given a known aggregator. When the coupling parameters are affected by uncertainty, robust NE need to be computed. We consider a scenario where players' weight in the aggregator is unknown, making the aggregator kind of "a black box". We pursue a suitable learning approach to estimate the unknown aggregator by proposing an inverse variational inequality-based relationship. We then utilize the counterpart to reconstruct the game and obtain first-order conditions for robust NE in the worst case. Furthermore, we characterize the generalization property of the learning methodology via an upper bound on the violation probability. Simulation experiments show the effectiveness of the proposed inverse learning approach.
Paper Structure (14 sections, 3 theorems, 40 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 3 theorems, 40 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Game Model with Uncertainty
  4. Robustness of Nash Equilibrium
  5. Problem Statement
  6. Inverse Learning
  7. Inverse VI-based Relationship
  8. Data-driven Optimization
  9. GNE Robustness
  10. Generalization Guarantee
  11. Violation Probability
  12. Generalization Bound
  13. Numerical Evaluation
  14. Conclusions

Key Result

Theorem 1

Consider the deterministic game $\mathscr G_{\bm \alpha}$ under a fixed parameter $\bm \alpha$. Under Assumption ass, $\bm x^*_{\bm \alpha}$ is a vGNE of $\mathscr G_{\bm \alpha}$ if and only if there exists a scalar $\gamma\leq 0$ such that

Figures (3)

  • Figure 1: Learning performance with data amounts $M=4$.
  • Figure 2: Convergence for seeking rGNE.
  • Figure 3: Learning performance with different data sizes.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Remark 2
  • Example 1
  • Theorem 2
  • Definition 4
  • Remark 3
  • ...and 2 more