Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control

TL;DR

This paper introduces the Deep BSDE-ML method, a novel approach for solving linear FBSDEs in stochastic control, extending the Deep BSDE framework with a measurability loss to improve approximation of control policies.

Contribution

The paper proposes a modified Deep BSDE method with a new measurability loss, enabling better approximation of gradients in PDEs and facilitating model-free optimal control learning.

Findings

Measurability loss equals the expected mean squared error of the diffusion term.

Deep BSDE-ML extends the application to approximate PDE gradients.

Framework successfully learns robust feedback controllers with exploration noise.

Abstract

A modified Deep BSDE (backward differential equation) learning method with measurability loss, called Deep BSDE-ML method, is introduced in this paper to solve a kind of linear decoupled forward-backward stochastic differential equations (FBSDEs), which is encountered in the policy evaluation of learning the optimal feedback policies of a class of stochastic control problems. The measurability loss is characterized via the measurability of BSDE's state at the forward initial time, which differs from that related to terminal state of the known Deep BSDE method. Though the minima of the two loss functions are shown to be equal, this measurability loss is proved to be equal to the expected mean squared error between the true diffusion term of BSDE and its approximation. This crucial observation extends the application of the Deep BSDE method -- approximating the gradients of the solution of a partial differential equation (PDE) instead of the solution itself. Simultaneously, a learning-based framework is introduced to search an optimal feedback control of a deterministic nonlinear system. Specifically, by introducing Gaussian exploration noise, we are aiming to learn a robust optimal controller under this stochastic case. This reformulation sacrifices the optimality to some extent, but as suggested in reinforcement learning (RL) exploration noise is essential to enable the model-free learning.