Approximate Bayesian Computation of B\'ezier Simplices

TL;DR

This paper introduces a probabilistic extension of Bzier simplex fitting using approximate Bayesian computation with Wasserstein distance, improving robustness to noisy data in multi-objective optimization.

Contribution

It proposes a novel Wasserstein ABC algorithm for Bzier simplex fitting, addressing overfitting issues with noisy samples and analyzing its convergence properties.

Findings

The new algorithm converges with finite samples.

It outperforms deterministic methods on noisy data.

Experimental results validate its effectiveness.

Abstract

B\'ezier simplex fitting algorithms have been recently proposed to approximate the Pareto set/front of multi-objective continuous optimization problems. These new methods have shown to be successful at approximating various shapes of Pareto sets/fronts when sample points exactly lie on the Pareto set/front. However, if the sample points scatter away from the Pareto set/front, those methods often likely suffer from over-fitting. To overcome this issue, in this paper, we extend the B\'ezier simplex model to a probabilistic one and propose a new learning algorithm of it, which falls into the framework of approximate Bayesian computation (ABC) based on the Wasserstein distance. We also study the convergence property of the Wasserstein ABC algorithm. An extensive experimental evaluation on publicly available problem instances shows that the new algorithm converges on a finite sample. Moreover, it outperforms the deterministic fitting methods on noisy instances.