Representation Policy Iteration

TL;DR

This paper introduces a theoretically rigorous framework for automatically generating basis functions for value function approximation in large MDPs, leveraging Riemannian geometry and spectral analysis to improve policy learning.

Contribution

It presents a novel, coordinate-free method for creating basis functions using manifold theory, enabling automatic learning of representations and policies in MDPs.

Findings

RPI outperforms handcoded basis functions in experiments

Basis functions reflect the topology of the state space

Framework is compatible with existing approximate MDP solvers

Abstract

This paper addresses a fundamental issue central to approximation methods for solving large Markov decision processes (MDPs): how to automatically learn the underlying representation for value function approximation? A novel theoretically rigorous framework is proposed that automatically generates geometrically customized orthonormal sets of basis functions, which can be used with any approximate MDP solver like least squares policy iteration (LSPI). The key innovation is a coordinate-free representation of value functions, using the theory of smooth functions on a Riemannian manifold. Hodge theory yields a constructive method for generating basis functions for approximating value functions based on the eigenfunctions of the self-adjoint (Laplace-Beltrami) operator on manifolds. In effect, this approach performs a global Fourier analysis on the state space graph to approximate value functions, where the basis functions reflect the largescale topology of the underlying state space. A new class of algorithms called Representation Policy Iteration (RPI) are presented that automatically learn both basis functions and approximately optimal policies. Illustrative experiments compare the performance of RPI with that of LSPI using two handcoded basis functions (RBF and polynomial state encodings).