Kullback-Leibler control for discrete-time nonlinear systems on continuous spaces

TL;DR

This paper reformulates KL control for continuous spaces to avoid unrealistic assumptions, enabling efficient numerical algorithms like Monte Carlo methods for nonlinear optimal control without losing optimality.

Contribution

It introduces a new formulation of KL control for continuous spaces that eliminates the need for full controllability assumptions, allowing practical computation.

Findings

Reformulated KL control does not require full controllability.

Monte Carlo methods can compute the value function efficiently.

The new approach maintains optimality without approximation.

Abstract

Kullback-Leibler (KL) control enables efficient numerical methods for nonlinear optimal control problems. The crucial assumption of KL control is the full controllability of the transition distribution. However, this assumption is often violated when the dynamics evolves in a continuous space. Consequently, applying KL control to problems with continuous spaces requires some approximation, which leads to the lost of the optimality. To avoid such approximation, in this paper, we reformulate the KL control problem for continuous spaces so that it does not require unrealistic assumptions. The key difference between the original and reformulated KL control is that the former measures the control effort by KL divergence between controlled and uncontrolled transition distributions while the latter replaces the uncontrolled transition by a noise-driven transition. We show that the reformulated KL control admits efficient numerical algorithms like the original one without unreasonable assumptions. Specifically, the associated value function can be computed by using a Monte Carlo method based on its path integral representation.