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A regularized variance-reduced modified extragradient method for stochastic hierarchical games

Shisheng Cui, Uday V. Shanbhag, Mathias Staudigl

TL;DR

The paper addresses stochastic N-player hierarchical games where each leader’s objective depends on an implicit follower equilibrium and an expectation-valued term. It introduces VRHGS, a two-loop, variance-reduced forward-backward-forward method that combines smoothing of the implicit map, iterative regularization, and stochastic variance reduction to obtain convergence to a least-norm hierarchical equilibrium and an O(1/T) rate for the ergodic gap. The results hold both for exact and inexact lower-level solves, and are supported by a detailed power-market-inspired model with virtual power plants to illustrate applicability. The approach reduces projection burdens and extends to general probability spaces, offering a scalable, provably convergent tool for large-scale stochastic hierarchical optimization and game-theoretic learning in energy systems and beyond.

Abstract

We consider an N-player hierarchical game in which the i-th player's objective comprises of an expectation-valued term, parametrized by rival decisions, and a hierarchical term. Such a framework allows for capturing a broad range of stochastic hierarchical optimization problems, Stackelberg equilibrium problems, and leader-follower games. We develop an iteratively regularized and smoothed variance-reduced modified extragradient framework for iteratively approaching hierarchical equilibria in a stochastic setting. We equip our analysis with rate statements, complexity guarantees, and almost-sure convergence results. We then extend these statements to settings where the lower-level problem is solved inexactly and provide the corresponding rate and complexity statements. Our model framework encompasses many game theoretic equilibrium problems studied in the context of power markets. We present a realistic application to the virtual power plants, emphasizing the role of hierarchical decision making and regularization.

A regularized variance-reduced modified extragradient method for stochastic hierarchical games

TL;DR

The paper addresses stochastic N-player hierarchical games where each leader’s objective depends on an implicit follower equilibrium and an expectation-valued term. It introduces VRHGS, a two-loop, variance-reduced forward-backward-forward method that combines smoothing of the implicit map, iterative regularization, and stochastic variance reduction to obtain convergence to a least-norm hierarchical equilibrium and an O(1/T) rate for the ergodic gap. The results hold both for exact and inexact lower-level solves, and are supported by a detailed power-market-inspired model with virtual power plants to illustrate applicability. The approach reduces projection burdens and extends to general probability spaces, offering a scalable, provably convergent tool for large-scale stochastic hierarchical optimization and game-theoretic learning in energy systems and beyond.

Abstract

We consider an N-player hierarchical game in which the i-th player's objective comprises of an expectation-valued term, parametrized by rival decisions, and a hierarchical term. Such a framework allows for capturing a broad range of stochastic hierarchical optimization problems, Stackelberg equilibrium problems, and leader-follower games. We develop an iteratively regularized and smoothed variance-reduced modified extragradient framework for iteratively approaching hierarchical equilibria in a stochastic setting. We equip our analysis with rate statements, complexity guarantees, and almost-sure convergence results. We then extend these statements to settings where the lower-level problem is solved inexactly and provide the corresponding rate and complexity statements. Our model framework encompasses many game theoretic equilibrium problems studied in the context of power markets. We present a realistic application to the virtual power plants, emphasizing the role of hierarchical decision making and regularization.
Paper Structure (27 sections, 12 theorems, 128 equations, 4 algorithms)

This paper contains 27 sections, 12 theorems, 128 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions and related work
  3. Hierarchical optimization, games and uncertain generalizations.
  4. Zeroth-order optimization, smoothing, and regularization.
  5. Variance Reduction.
  6. Preliminaries
  7. A variance reduced forward-backward-forward algorithm for hierarchical games
  8. Iterative Regularization methods
  9. The Algorithm
  10. The outer loop
  11. The inner loop
  12. Discussion
  13. Main Results
  14. Hierarchical games in power markets
  15. Proof of the Main Theorem
  16. ...and 12 more sections

Key Result

Lemma 2.2

Let Assumptions ass:LLunique-ass:standing hold. Then $h_{i}(\mathbf{x}_{i})=g_{i}(\mathbf{x}_{i},\mathbf{y}_{i}(\mathbf{x}_{i}))$ is $L_{h_{i}}$-Lipschitz continuous on $\mathcal{X}_{i}$ and directionally differentiable.

Theorems & Definitions (21)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 3.1
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • Lemma 6.3
  • ...and 11 more