Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms

TL;DR

This paper analyzes the limitations of max-plus approximation methods in high-dimensional control problems, providing theoretical error bounds and proposing improved algorithms to mitigate the curse of dimensionality.

Contribution

It introduces a new theoretical framework linking approximation problems to combinatorial optimization and refines existing methods for better accuracy at similar computational costs.

Findings

Theoretical error estimates for basis function approximation

Connection of approximation problems to facility location and k-center problems

Enhanced pruning algorithms for improved accuracy

Abstract

Max-plus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (max-plus "basis functions") taken from a prescribed dictionary. We study several variants of this approximation problem, which we show to be continuous versions of the facility location and $k$-center combinatorial optimization problems, in which the connection costs arise from a Bregman distance. We give theoretical error estimates, quantifying the number of basis functions needed to reach a prescribed accuracy. We derive from our approach a refinement of the curse of dimensionality free method introduced previously by McEneaney, with a higher accuracy for a comparable computational cost.