A Study of Policy Gradient on a Class of Exactly Solvable Models

TL;DR

This paper analytically studies the behavior of policy gradient methods in a special class of solvable POMDPs, revealing insights into convergence and landscape complexity using novel environments and random walk theory.

Contribution

It introduces a new analytical framework for understanding policy gradient dynamics in exactly solvable POMDPs, leveraging affine Weyl groups and novel environments.

Findings

Policy gradient can converge to different local maxima.

The value distribution influences policy parameter evolution.

Analytical computation of the policy landscape in POMDPs is possible.

Abstract

Policy gradient methods are extensively used in reinforcement learning as a way to optimize expected return. In this paper, we explore the evolution of the policy parameters, for a special class of exactly solvable POMDPs, as a continuous-state Markov chain, whose transition probabilities are determined by the gradient of the distribution of the policy's value. Our approach relies heavily on random walk theory, specifically on affine Weyl groups. We construct a class of novel partially observable environments with controllable exploration difficulty, in which the value distribution, and hence the policy parameter evolution, can be derived analytically. Using these environments, we analyze the probabilistic convergence of policy gradient to different local maxima of the value function. To our knowledge, this is the first approach developed to analytically compute the landscape of policy gradient in POMDPs for a class of such environments, leading to interesting insights into the difficulty of this problem.