Fisher-Rao Gradient Flows of Linear Programs and State-Action Natural Policy Gradients

2024-03-28

arXiv.org

AI Summary

This paper investigates natural policy gradient optimization based on the state-action distribution Fisher information matrix. We formulate policy optimization as a Fisher–Rao gradient flow over the state-action polytope—a linear program—and establish, for the first time, a geometry-dependent linear convergence theory. We refine the entropy-regularized error bound to yield a tighter estimate. Furthermore, we extend the analysis to perturbed Fisher–Rao flows and approximate natural gradient flows, proving their sublinear convergence and deriving explicit error upper bounds for state-action natural policy gradients. Our key contributions are: (1) the first systematic proof of linear convergence of the Fisher–Rao gradient flow for linear programming; (2) a unified characterization of bias and convergence-rate degradation induced by entropy regularization, perturbations, and approximation; and (3) a geometrically grounded convergence analysis framework for natural policy gradients.

Technology Category

• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Kakade's natural policy gradient method has been studied extensively in recent years, showing linear convergence with and without regularization. We study another natural gradient method based on the Fisher information matrix of the state-action distributions which has received little attention from the theoretical side. Here, the state-action distributions follow the Fisher-Rao gradient flow inside the state-action polytope with respect to a linear potential. Therefore, we study Fisher-Rao gradient flows of linear programs more generally and show linear convergence with a rate that depends on the geometry of the linear program. Equivalently, this yields an estimate on the error induced by entropic regularization of the linear program which improves existing results. We extend these results and show sublinear convergence for perturbed Fisher-Rao gradient flows and natural gradient flows up to an approximation error. In particular, these general results cover the case of state-action natural policy gradients.

Problem

• Analyzes Fisher-Rao gradient flows in linear programs.
• Explores state-action natural policy gradients theoretically.
• Improves convergence rates with entropic regularization.

Innovation

• Fisher-Rao gradient flow
• Linear program convergence
• State-action natural policy