Geometry and convergence of natural policy gradient methods

TL;DR

This paper analyzes the convergence properties of natural policy gradient methods in reinforcement learning, revealing their geometric structure and providing guarantees for various regularizations and geometries.

Contribution

It introduces a geometric framework for understanding NPG convergence, deriving global and local rates, and connecting discrete NPG to inexact Newton methods.

Findings

Global convergence guarantees for several NPG variants.

Linear convergence rates for regularized NPG flows.

Local quadratic convergence for regularized NPG as inexact Newton methods.

Abstract

We study the convergence of several natural policy gradient (NPG) methods in infinite-horizon discounted Markov decision processes with regular policy parametrizations. For a variety of NPGs and reward functions we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows with the metrics proposed by Kakade and Morimura and co-authors by observing that these arise from the Hessian geometries of conditional entropy and entropy respectively. Further, we obtain sublinear convergence rates for Hessian geometries arising from other convex functions like log-barriers. Finally, we interpret the discrete-time NPG methods with regularized rewards as inexact Newton methods if the NPG is defined with respect to the Hessian geometry of the regularizer. This yields local quadratic convergence rates of these methods for step size equal to the penalization strength.