Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic
Takehiro Tottori1, Tetsuya J Kobayashi1,2,3,4
1Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8654, Japan.
Entropy (Basel, Switzerland)
|February 25, 2023
Summary
This study introduces a new method for memory-limited partially observable stochastic control (ML-POSC). The forward-backward sweep method (FBSM) is proven to converge for ML-POSC problems, unlike in other control types.
Area of Science:
- Control Theory
- Stochastic Processes
- Mathematical Optimization
Background:
- Memory-limited partially observable stochastic control (ML-POSC) presents challenges in optimal control under incomplete information and memory constraints.
- Solving ML-POSC typically requires addressing coupled forward Fokker-Planck (FP) and backward Hamilton-Jacobi-Bellman (HJB) equations.
Purpose of the Study:
- To interpret the HJB-FP equation system using Pontryagin's minimum principle in the probability density function space.
- To propose and validate the forward-backward sweep method (FBSM) for solving ML-POSC problems.
Main Methods:
- Interpreting the HJB-FP system through Pontryagin's minimum principle.
- Applying the forward-backward sweep method (FBSM), which iteratively solves the forward FP and backward HJB equations.
Main Results:
- Demonstrated that the HJB-FP system in ML-POSC can be viewed through the lens of Pontryagin's minimum principle.
- Established the guaranteed convergence of FBSM for ML-POSC, a significant improvement over its performance in other control domains.
Conclusions:
- The forward-backward sweep method (FBSM) is a convergent algorithm for memory-limited partially observable stochastic control.
- The limited coupling in ML-POSC's HJB-FP equations is key to ensuring FBSM convergence.
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