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Solving Decision-Dependent Games by Learning from Feedback

Killian Wood, Ahmed Zamzam, Emiliano Dall'Anese

TL;DR

The paper addresses solving stochastic Nash equilibria when data distributions are decision-dependent and unknown. It introduces a learning-based plug-in framework that first estimates a distributional map $D_i$ from responses and then optimizes using the learned map in a distributed gradient setting, yielding a Nash equilibrium of the approximate game. Theoretical guarantees include high-probability ERM bounds, uniform gradient bounds for map estimation, and linear convergence of the distributed gradient method to a neighborhood of the equilibrium, with decaying step-sizes enabling convergence to the true point. Numerical experiments on an electric vehicle charging market with real data demonstrate the practical viability and impact of the approach for decision-dependent data in multi-agent settings.

Abstract

This paper tackles the problem of solving stochastic optimization problems with a decision-dependent distribution in the setting of stochastic strongly-monotone games and when the distributional dependence is unknown. A two-stage approach is proposed, which initially involves estimating the distributional dependence on decision variables, and subsequently optimizing over the estimated distributional map. The paper presents guarantees for the approximation of the cost of each agent. Furthermore, a stochastic gradient-based algorithm is developed and analyzed for finding the Nash equilibrium in a distributed fashion. Numerical simulations are provided for a novel electric vehicle charging market formulation using real-world data.

Solving Decision-Dependent Games by Learning from Feedback

TL;DR

The paper addresses solving stochastic Nash equilibria when data distributions are decision-dependent and unknown. It introduces a learning-based plug-in framework that first estimates a distributional map from responses and then optimizes using the learned map in a distributed gradient setting, yielding a Nash equilibrium of the approximate game. Theoretical guarantees include high-probability ERM bounds, uniform gradient bounds for map estimation, and linear convergence of the distributed gradient method to a neighborhood of the equilibrium, with decaying step-sizes enabling convergence to the true point. Numerical experiments on an electric vehicle charging market with real data demonstrate the practical viability and impact of the approach for decision-dependent data in multi-agent settings.

Abstract

This paper tackles the problem of solving stochastic optimization problems with a decision-dependent distribution in the setting of stochastic strongly-monotone games and when the distributional dependence is unknown. A two-stage approach is proposed, which initially involves estimating the distributional dependence on decision variables, and subsequently optimizing over the estimated distributional map. The paper presents guarantees for the approximation of the cost of each agent. Furthermore, a stochastic gradient-based algorithm is developed and analyzed for finding the Nash equilibrium in a distributed fashion. Numerical simulations are provided for a novel electric vehicle charging market formulation using real-world data.
Paper Structure (22 sections, 10 theorems, 83 equations, 3 figures, 1 algorithm)

This paper contains 22 sections, 10 theorems, 83 equations, 3 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. CONTRIBUTIONS
  4. ORGANIZATION
  5. Notation and Preliminaries
  6. Probability measures
  7. Games
  8. Monotonicity in Decision-Dependent Games
  9. Learning-based Decision-Dependent Games
  10. Parameter Estimation for Regular Problems
  11. Bounding the Approximation Error
  12. Solving Strongly-monotone Decision-dependent Games
  13. Distributed Gradient-based Method
  14. Numerical Experiments on Electric Vehicle Charging
  15. CONCLUSIONS
  16. ...and 7 more sections

Key Result

theorem 1

(Strong Monotonicity, narang2022learning) Suppose that, and that for all $i\in[n]$, Set $\kappa = \sqrt{\sum_{i=1}^{n}( \frac{\gamma_{i}L_{i}}{\lambda})^{2}}$. Then if $\kappa < 1/2$, $x\mapsto G(x)$ is $\alpha = (1-2\kappa)\lambda$-strongly monotone. $\Box$

Figures (3)

  • Figure 1: Communication structure allows agents to interact with the system in square by sending decision $x_{i}$. After deploying, agents can receive feedback from the system in the form of other agents decisions $x_{-i}$ and data $z_{i}$.
  • Figure 2: Standardized demand data for six medium demand EVCS's consisting of either 2 or 6 ports and port power values of 50, 150, and 350 kWh. Standardization maps raw demand instances to instances of demand that are deviations from the average at each station.
  • Figure 3: Expected error curve and confidence interval for regularized stochastic gradient descent with decaying step size for a location-scale model.

Theorems & Definitions (13)

  • theorem 1
  • definition 1
  • lemma 1
  • theorem 2
  • proposition 1
  • definition 2
  • definition 3
  • theorem 3
  • corollary 1
  • lemma 2
  • ...and 3 more