Max-Plus Matching Pursuit for Deterministic Markov Decision Processes

TL;DR

This paper introduces a max-plus algebra-based matching pursuit method for approximating value functions in deterministic MDPs, reducing complexity and enabling adaptive basis selection to address high-dimensional problems.

Contribution

It develops a novel max-plus algebra framework for value iteration, including adaptive basis methods inspired by signal processing, with empirical success on low-dimensional control problems.

Findings

Complexity depends on covering numbers, not state count

Adaptive basis methods improve approximation in factored state-spaces

Empirical results show effectiveness on simple deterministic MDPs

Abstract

We consider deterministic Markov decision processes (MDPs) and apply max-plus algebra tools to approximate the value iteration algorithm by a smaller-dimensional iteration based on a representation on dictionaries of value functions. The setup naturally leads to novel theoretical results which are simply formulated due to the max-plus algebra structure. For example, when considering a fixed (non adaptive) finite basis, the computational complexity of approximating the optimal value function is not directly related to the number of states, but to notions of covering numbers of the state space. In order to break the curse of dimensionality in factored state-spaces, we consider adaptive basis that can adapt to particular problems leading to an algorithm similar to matching pursuit from signal processing. They currently come with no theoretical guarantees but work empirically well on simple deterministic MDPs derived from low-dimensional continuous control problems. We focus primarily on deterministic MDPs but note that the framework can be applied to all MDPs by considering measure-based formulations.