# Entropic Regularization of Markov Decision Processes

## TL;DR

This paper introduces a unified entropic regularization framework for Markov decision processes using $f$-divergences, enabling stable policy learning and providing insights into different divergence choices' effects on reinforcement learning.

## Contribution

It generalizes KL-based regularization to a broader family of $f$-divergences, offering closed-form policy updates and a unified actor-critic perspective.

## Key findings

- Closed-form policy improvement for various divergences.
- Connection between least-squares value estimation and Pearson $ ext{chi}^2$-divergence.
- Asymptotic analysis of divergence effects on standard RL problems.

## Abstract

An optimal feedback controller for a given Markov decision process (MDP) can in principle be synthesized by value or policy iteration. However, if the system dynamics and the reward function are unknown, a learning agent must discover an optimal controller via direct interaction with the environment. Such interactive data gathering commonly leads to divergence towards dangerous or uninformative regions of the state space unless additional regularization measures are taken. Prior works proposed bounding the information loss measured by the Kullback-Leibler (KL) divergence at every policy improvement step to eliminate instability in the learning dynamics. In this paper, we consider a broader family of $f$-divergences, and more concretely $\alpha$-divergences, which inherit the beneficial property of providing the policy improvement step in closed form at the same time yielding a corresponding dual objective for policy evaluation. Such entropic proximal policy optimization view gives a unified perspective on compatible actor-critic architectures. In particular, common least-squares value function estimation coupled with advantage-weighted maximum likelihood policy improvement is shown to correspond to the Pearson $\chi^2$-divergence penalty. Other actor-critic pairs arise for various choices of the penalty-generating function $f$. On a concrete instantiation of our framework with the $\alpha$-divergence, we carry out asymptotic analysis of the solutions for different values of $\alpha$ and demonstrate the effects of the divergence function choice on common standard reinforcement learning problems.