Geometry and convergence of natural policy gradient methods
This provides theoretical foundations for policy optimization in reinforcement learning, offering convergence insights for practitioners, though it is incremental as it builds on existing NPG frameworks.
The paper tackles the convergence analysis of natural policy gradient (NPG) methods in infinite-horizon discounted Markov decision processes by showing that their trajectories correspond to gradient flows in Hessian geometries, leading to global convergence guarantees and rates, including linear convergence for specific metrics and sublinear rates for others.
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.