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A Method to Improve the Performance of Reinforcement Learning Based on the Y Operator for a Class of Stochastic Differential Equation-Based Child-Mother Systems

Cheng Yin, Yi Chen

TL;DR

The paper tackles optimal control for systems with stochastic dynamics described by $\\mathcal{SDE}$s, focusing on a child-mother system and the challenge of incorporating stochasticity into value-function estimation. It introduces the $\\mathcal{Y}$ operator, proven equivalent to the Itô generator $\\mathcal{A}$, to transform the time derivative of value-function functionals into partial derivatives of the drift and diffusion terms, enabling the YORL framework. The method integrates the operator into a Critic loss and uses PPO-style Actor updates, augmented by a data-driven NSDE calibration step. Across linear and nonlinear NSDE experiments, YORL outperforms traditional TSRL in convergence and final reward, with flexibility in activation choices and applicability to offline/IRL scenarios, highlighting practical impact for stochastic control with neural RL.

Abstract

This paper introduces a novel operator, termed the Y operator, to elevate control performance in Actor-Critic(AC) based reinforcement learning for systems governed by stochastic differential equations(SDEs). The Y operator ingeniously integrates the stochasticity of a class of child-mother system into the Critic network's loss function, yielding substantial advancements in the control performance of RL algorithms.Additionally, the Y operator elegantly reformulates the challenge of solving partial differential equations for the state-value function into a parallel problem for the drift and diffusion functions within the system's SDEs.A rigorous mathematical proof confirms the operator's validity.This transformation enables the Y Operator-based Reinforcement Learning(YORL) framework to efficiently tackle optimal control problems in both model-based and data-driven systems.The superiority of YORL is demonstrated through linear and nonlinear numerical examples showing its enhanced performance over existing methods post convergence.

A Method to Improve the Performance of Reinforcement Learning Based on the Y Operator for a Class of Stochastic Differential Equation-Based Child-Mother Systems

TL;DR

The paper tackles optimal control for systems with stochastic dynamics described by s, focusing on a child-mother system and the challenge of incorporating stochasticity into value-function estimation. It introduces the operator, proven equivalent to the Itô generator , to transform the time derivative of value-function functionals into partial derivatives of the drift and diffusion terms, enabling the YORL framework. The method integrates the operator into a Critic loss and uses PPO-style Actor updates, augmented by a data-driven NSDE calibration step. Across linear and nonlinear NSDE experiments, YORL outperforms traditional TSRL in convergence and final reward, with flexibility in activation choices and applicability to offline/IRL scenarios, highlighting practical impact for stochastic control with neural RL.

Abstract

This paper introduces a novel operator, termed the Y operator, to elevate control performance in Actor-Critic(AC) based reinforcement learning for systems governed by stochastic differential equations(SDEs). The Y operator ingeniously integrates the stochasticity of a class of child-mother system into the Critic network's loss function, yielding substantial advancements in the control performance of RL algorithms.Additionally, the Y operator elegantly reformulates the challenge of solving partial differential equations for the state-value function into a parallel problem for the drift and diffusion functions within the system's SDEs.A rigorous mathematical proof confirms the operator's validity.This transformation enables the Y Operator-based Reinforcement Learning(YORL) framework to efficiently tackle optimal control problems in both model-based and data-driven systems.The superiority of YORL is demonstrated through linear and nonlinear numerical examples showing its enhanced performance over existing methods post convergence.
Paper Structure (16 sections, 8 theorems, 80 equations, 2 figures, 1 table)

This paper contains 16 sections, 8 theorems, 80 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. THEORY
  4. SYSTEM MODELING FOR A CLASS OF CHILD-MOTHER SYSTEM
  5. A Class Of Child-Mother System Modeling
  6. Stochastic Differential Equation Calibration of Child-Mother System
  7. RL DESIGN BASED ON $\mathcal{Y}$ OPERATOR IN CHILD-MOTHER SYSTEM
  8. Critic Network Design
  9. Actor Network Design
  10. ILLUSTRATIVE EXAMPLES
  11. Linear Numerical Examples
  12. Nonlinear Numerical Examples
  13. CONCLUSIONS
  14. APPENDIX
  15. The Proof of Proposition \ref{['theory proposition1']}
  16. ...and 1 more sections

Key Result

Lemma 1

{evans2012introductionsection5.2: Existence and uniqueness of solution of stochastic differential equation} Support that $h:\mathbb{R}^n\times[0,T]\rightarrow\mathbb{R}^n$ and $H:\mathbb{R}^n\times[0,T]\rightarrow\mathbb{R}^{n\times m}$ are continuous and satisfy the following conditions: Let $X_0$ be any $\mathbb{R}^n$-valued random variable such that and where $W(\cdot)$ is a given $m$-dimens

Figures (2)

  • Figure 1: In subfigure (a), the hidden layer dimension of both the Actor and Critic networks for both TSRL and YORL is $32$ and the activation function used is the Sigmoid function. In subfigure (b), the hidden layer dimension is $128$ and the activation function used is Sigmoid. In subfigure (c), the hidden layer dimension is $32$ and the activation function used is Relu. In subfigure (d), the hidden layer dimension is $32$ and the activation function used is tanh.
  • Figure 2: In subfigure (a), the hidden layer dimension of both the Actor and Critic networks for both TSRL and YORL is $32$ and the activation function used is the Sigmoid function. In subfigure (b), the hidden layer dimension is $128$ and the activation function used is Sigmoid. In subfigure (c), the hidden layer dimension is $32$ and the activation function used is Relu. In subfigure (d), the hidden layer dimension is $32$ and the activation function used is tanh.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Definition 5
  • Corollary 1
  • Corollary 2
  • Definition 6
  • ...and 7 more