Asymptotic Risk of Bezier Simplex Fitting

TL;DR

This paper analyzes the asymptotic risks of Bezier simplex fitting methods for modeling Pareto fronts in multi-objective optimization, deriving optimal sampling strategies and demonstrating their effectiveness through numerical experiments and applications.

Contribution

It provides the first asymptotic risk analysis of Bezier simplex fitting methods and identifies optimal sampling ratios, enhancing understanding of their performance.

Findings

Inductive skeleton fitting with optimal ratio has lower risk for degree less than three.

All-at-once fitting performs better for degree three Bezier simplexes.

Numerical verification confirms theoretical risk bounds.

Abstract

The Bezier simplex fitting is a novel data modeling technique which exploits geometric structures of data to approximate the Pareto front of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from each skeleton of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a whole. In this paper, we analyze the asymptotic risks of those B\'ezier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a Bezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a Bezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a Bezier simplex of degree three where the all-at-once fitting gets an advantage.