Performance guarantees for model-based Approximate Dynamic Programming in continuous spaces

TL;DR

This paper provides theoretical performance guarantees for model-based Approximate Dynamic Programming in continuous spaces, analyzing both value and Q-function formulations to improve understanding and computational efficiency.

Contribution

It offers new guarantees for fitting error and online performance, and introduces a simplified Q-function formulation for better computational efficiency.

Findings

Guarantees for fitting error and online policy performance.

A simplified Q-function formulation reduces computational complexity.

Analysis of iterated policies and inequalities enhances theoretical understanding.

Abstract

We study both the value function and Q-function formulation of the Linear Programming approach to Approximate Dynamic Programming. The approach is model-based and optimizes over a restricted function space to approximate the value function or Q-function. Working in the discrete time, continuous space setting, we provide guarantees for the fitting error and online performance of the policy. In particular, the online performance guarantee is obtained by analyzing an iterated version of the greedy policy, and the fitting error guarantee by analyzing an iterated version of the Bellman inequality. These guarantees complement the existing bounds that appear in the literature. The Q-function formulation offers benefits, for example, in decentralized controller design, however it can lead to computationally demanding optimization problems. To alleviate this drawback, we provide a condition that simplifies the formulation, resulting in improved computational times.