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DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems

Michael Ludkovski, Changgen Xie, Zimu Zhu

TL;DR

This work tackles numerical resolution of continuous-time Principal-Agent problems by introducing DeepPAAC, a deep learning variant of the Deep Galerkin Method organized as an Actor-Critic algorithm to handle implicit Hamiltonians. The method uses separate neural networks for the value function and the optimal feedback control, minimizing PDE residuals and embedding a terminal constraint through a correction term. The authors validate DeepPAAC across five multidimensional case studies, including constrained contracts and explicit-solution benchmarks, demonstrating stability, accuracy, and faster convergence relative to prior DGM-based approaches. This approach broadens the computational toolkit for complex PA models and highlights the potential of neural PDE solvers for high-dimensional stochastic control problems.

Abstract

We consider numerical resolution of principal-agent (PA) problems in continuous time. We formulate a generic PA model with continuous and lump payments and a multi-dimensional strategy of the agent. To tackle the resulting Hamilton-Jacobi-Bellman equation with an implicit Hamiltonian we develop a novel deep learning method: the Deep Principal-Agent Actor Critic (DeepPAAC) Actor-Critic algorithm. DeepPAAC is able to handle multi-dimensional states and controls, as well as constraints. We investigate the role of the neural network architecture, training designs, loss functions, etc. on the convergence of the solver, presenting five different case studies.

DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems

TL;DR

This work tackles numerical resolution of continuous-time Principal-Agent problems by introducing DeepPAAC, a deep learning variant of the Deep Galerkin Method organized as an Actor-Critic algorithm to handle implicit Hamiltonians. The method uses separate neural networks for the value function and the optimal feedback control, minimizing PDE residuals and embedding a terminal constraint through a correction term. The authors validate DeepPAAC across five multidimensional case studies, including constrained contracts and explicit-solution benchmarks, demonstrating stability, accuracy, and faster convergence relative to prior DGM-based approaches. This approach broadens the computational toolkit for complex PA models and highlights the potential of neural PDE solvers for high-dimensional stochastic control problems.

Abstract

We consider numerical resolution of principal-agent (PA) problems in continuous time. We formulate a generic PA model with continuous and lump payments and a multi-dimensional strategy of the agent. To tackle the resulting Hamilton-Jacobi-Bellman equation with an implicit Hamiltonian we develop a novel deep learning method: the Deep Principal-Agent Actor Critic (DeepPAAC) Actor-Critic algorithm. DeepPAAC is able to handle multi-dimensional states and controls, as well as constraints. We investigate the role of the neural network architecture, training designs, loss functions, etc. on the convergence of the solver, presenting five different case studies.

Paper Structure

This paper contains 18 sections, 3 theorems, 71 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. HJB Equations and Critic Actor Algorithm
  4. Policy Iteration
  5. Principal-Agent Actor Critic Algorithm
  6. Implementation
  7. Stopping Criteria
  8. Neural Network Architecture
  9. Training Designs
  10. A toy example: The Holmström--Milgrom Model
  11. Multidimensional Case Studies
  12. A Solely Continuous Payment
  13. DeepPAAC Variants
  14. PA problem with Constrained Contracts
  15. Two-dimensional Problem with Scalar Control and Explicit Solution
  16. ...and 3 more sections

Key Result

Proposition 3.1

The Principal's value function $V^P(t,x)$ can be characterized as the solution of the following HJB equation: When $\Phi^P(x)=-\exp(-\gamma_P x), \gamma_P >0$ is of exponential type, we have the explicit solution: and Agent's optimal control is constant The optimal terminal lump-sum contract is

Figures (9)

  • Figure 1: Top: NN architecture for value function. Below: NN architecture for control
  • Figure 2: Estimated optimal control ${a}(t,x)$ obtained from the DeepPAAC algorithm for different exponential mixture weights $\lambda \in (0,1)$.
  • Figure 3: Left:${a}(t,x;0)$ for $\lambda=0$. Right:${a}(t,x;0.5)$ for $\lambda=0.5$
  • Figure 4: Left (a): Contour plot of the value function error metrics $\bar{L}_{int}(\theta_n^V;t,w)$ at $n=300$ training steps. Right (b):$L_2$-norm and $L_\infty$-norm for $\bar{L}_{int}(\theta_n^V;\bar{t}^\cdot,\bar{w}^{\cdot})$ across $n$.
  • Figure 5: Validating convergence of the DeepPAAC loss functions. We show the $L_\infty$-norm for the value function HJB residual $\bar{L}_{int}( \theta^V_n; {\bar{t}}^\cdot, \bar{w}^\cdot)$ (left panel (a)) and the control loss criterion $\bar{L}_{ctrl}( \theta^V_n, \theta^u_n; {\bar{t}}^\cdot, \bar{w}^\cdot)$ (right panel (b)) as a function of step $n$, for the case study from Section \ref{['subsec: multi control-one dim spatial']}. Five runs of the DeepPAAC scheme with a single control NN.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • Remark 1
  • Proposition 4.1
  • Proposition 4.2