Exploratory Randomization for Discrete-Time Linear Exponential Quadratic Gaussian (LEQG) Problem

TL;DR

This paper explores the use of randomized control strategies in discrete-time LEQG problems, linking risk-sensitive control with entropy regularization through duality, and providing a solution via dynamic programming.

Contribution

It introduces a novel approach to incorporate exploration in LEQG control by randomization and connects it with entropy regularization using energy-entropy duality, offering a new justification for this technique.

Findings

Reduces LEQG to risk-neutral LQG with entropy regularization

Provides a dynamic programming solution approach

Justifies entropy regularization in randomized control contexts

Abstract

We investigate exploratory randomization for an extended linear-exponential-quadratic-Gaussian (LEQG) control problem in discrete time. This extended control problem is related to the structure of risk-sensitive investment management applications. We introduce exploration through a randomization of the control. Next, we apply the duality between free energy and relative entropy to reduce the LEQG problem to an equivalent risk-neutral LQG control problem with an entropy regularization term, see, e.g. Dai Pra et al. (1996), for which we present a solution approach based on Dynamic Programming. Our approach, based on the energy-entropy duality may also be considered as leading to a justification for the use, in the literature, of an entropy regularization when applying a randomized control.