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Actively learning equilibria in Nash games with misleading information

Barbara Franci, Filippo Fabiani, Alberto Bemporad

TL;DR

This work addresses learning equilibria in a Generalized Nash Equilibrium Problem when agents provide privacy-masked, potentially biased information. It introduces an active-learning framework where an external observer iteratively updates BR proxies via an inexact proximal scheme, selects informative query points by minimizing the distance to these proxies, and collects noisy best-response samples to refine the proxies. Under standard stochastic-optimization assumptions, the method guarantees convergence of the proxies to exact BR mappings and of the predicted equilibrium to the true GNE, despite misleading data. The approach is demonstrated on a smart-grid charging case study, highlighting practical considerations such as batch size, synthetic sampling, and the advantages of affine proxies for tractable, convergent learning. The results indicate a viable path for indirect control of multi-agent systems where privacy and noise complicate direct estimation of agents' costs and responses.

Abstract

We develop a scheme based on active learning to compute equilibria in a generalized Nash equilibrium problem (GNEP). Specifically, an external observer (or entity), with little knowledge on the multi-agent process at hand, collects sensible data by probing the agents' best-response (BR) mappings, which are then used to recursively update local parametric estimates of these mappings. Unlike [1], we consider the realistic case in which the agents share corrupted information with the external entity for, e.g., protecting their privacy. Inspired by a popular approach in stochastic optimization, we endow the external observer with an inexact proximal scheme for updating the local BR proxies. This technique will prove key to establishing the convergence of our scheme under standard assumptions, thereby enabling the external observer to predict an equilibrium strategy even when relying on masked information.

Actively learning equilibria in Nash games with misleading information

TL;DR

This work addresses learning equilibria in a Generalized Nash Equilibrium Problem when agents provide privacy-masked, potentially biased information. It introduces an active-learning framework where an external observer iteratively updates BR proxies via an inexact proximal scheme, selects informative query points by minimizing the distance to these proxies, and collects noisy best-response samples to refine the proxies. Under standard stochastic-optimization assumptions, the method guarantees convergence of the proxies to exact BR mappings and of the predicted equilibrium to the true GNE, despite misleading data. The approach is demonstrated on a smart-grid charging case study, highlighting practical considerations such as batch size, synthetic sampling, and the advantages of affine proxies for tractable, convergent learning. The results indicate a viable path for indirect control of multi-agent systems where privacy and noise complicate direct estimation of agents' costs and responses.

Abstract

We develop a scheme based on active learning to compute equilibria in a generalized Nash equilibrium problem (GNEP). Specifically, an external observer (or entity), with little knowledge on the multi-agent process at hand, collects sensible data by probing the agents' best-response (BR) mappings, which are then used to recursively update local parametric estimates of these mappings. Unlike [1], we consider the realistic case in which the agents share corrupted information with the external entity for, e.g., protecting their privacy. Inspired by a popular approach in stochastic optimization, we endow the external observer with an inexact proximal scheme for updating the local BR proxies. This technique will prove key to establishing the convergence of our scheme under standard assumptions, thereby enabling the external observer to predict an equilibrium strategy even when relying on masked information.

Paper Structure

This paper contains 8 sections, 3 theorems, 12 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Active learning with misleading information
  4. Convergence analysis
  5. Implementation details and simulation results
  6. Practical considerations
  7. Case study: Indirect control of smart grids
  8. Conclusion

Key Result

Lemma 1

For all $i \in \mathcal{I}$, let $\{\theta_i^k\}_{k\in\mathbb{N}}$ be the sequence generated by eq_inexact_br in Algorithm alg:learning_embedded_BR. If $\lim_{k \to \infty} \mathbb{E}_{\mathbb{P}_i}[z_i^k] = \bar{x}_i$, $\lim_{k \to \infty} \hat{\boldsymbol{x}}^k_{-i} = \bar{\boldsymbol{x}}_{-i}$ so

Figures (2)

  • Figure 1: Relative distance between the computed by relying on misleading information through Algorithm \ref{['alg:learning_embedded_BR']} (i.e., $\boldsymbol{x}^{200}$), and the one obtained with noiseless data (i.e., $\boldsymbol{x}^\star$), averaged over $20$ numerical instances of \ref{['eq:single_prob_EV']}.
  • Figure 2: Relative distance sequence produced by Algorithm \ref{['alg:learning_embedded_BR']} (solid blue line) and by fabiani2024active exploiting noisy data (solid red line), averaged over $20$ numerical instances. The shaded colored areas represent the standard deviation over the different numerical trials, while the shaded black region corresponds to the random initialization of both procedures.

Theorems & Definitions (8)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof