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Neural Operators Can Play Dynamic Stackelberg Games

Guillermo Alvarez, Ibrahim Ekren, Anastasis Kratsios, Xuwei Yang

TL;DR

The main result is obtained using the universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.

Abstract

Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.

Neural Operators Can Play Dynamic Stackelberg Games

TL;DR

The main result is obtained using the universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.

Abstract

Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.

Paper Structure

This paper contains 28 sections, 11 theorems, 158 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organization of Paper
  3. Related Literature
  4. Neural Operators
  5. Preliminaries
  6. Predictable Processes
  7. Examples of Compact Sets of Square-Integrable Controls
  8. The Dynamic Stackelberg Game
  9. Neural Operators
  10. The Attentional Neural Operator Model
  11. Link Between Our Decoder, Attention, and Transformers
  12. Main Results
  13. An Unsupervised Objective Function - Bypassing Computing U
  14. Rates For Perturbations of Closed-Loop Solutions to Linearized Game
  15. Overview of Proof for Main Results
  16. ...and 13 more sections

Key Result

Theorem 4.1

Under Assumptions assm:LipfsigLg and assm:continuity, for each compact ${\mathcal{K}}_0\subseteq {\mathcal{U}}_0$, and each $\varepsilon>0$ there is an encoding dimension $d\in \mathbb{N}_+$ and a neural operator $\hat{U}\in \mathcal{NO}:{\mathcal{U}}_0\to{\mathcal{U}}_1$ satisfying

Figures (2)

  • Figure 1: Attentional Neural Operator Workflow: Our attentionalneural operator model maps controls $u_{\cdot}$ to square-integrable $\mathbb{F}$-adapted processes $\hat{U}(u_{\cdot})$ in three phases. First, the (input) control is linearly projected onto the wavelet-like (in time) Wiener Chaos-like (in space) orthonormal basis of $\mathcal{H}_T^2$. Next, the basis coefficients are transformed by a feedforward neural network (MLP). Lastly, the basis coefficients are used to identify extremal points in a simplex in $\mathcal{H}_T^2$ and the outputs of the MLP are used to parameterize a prediction in its relative interior.
  • Figure 2: The ellipsoidal compact set $K$ of Example \ref{['ex:perturbations__lin_prob_control']}.

Theorems & Definitions (39)

  • Example 1: Compactness Via Regularity of the Malliavin Derivative
  • Example 2: Compactness Via Continuously Differentiable Martingale Controls
  • Example 3: Deterministic Hölder Continuous Controls
  • Example 4: Conditioned Lipschitz Perturbations of Random Variables at Terminal Time
  • Remark 3.1: Alternative Proofs of Compactness Directly Via Example \ref{['ex:CompactnessviaMalliavin']}
  • Example 5: Finite Sets of Controls
  • Definition 3.3: Stackelberg Equilibrium
  • Remark 3.5
  • Example 6: KidgerLyons-Type "Standard" Trainable Functions
  • Example 7: Super-Expressive Activation with Neuron-Specific Skip-Connection
  • ...and 29 more