Asymptotic Risk of Bezier Simplex Fitting
This work provides theoretical insights for researchers in multi-objective optimization, offering incremental improvements in risk analysis for specific Bezier simplex fitting methods.
The paper analyzes the asymptotic risks of two Bezier simplex fitting methods for approximating Pareto fronts in multi-objective optimization, deriving an optimal subsample ratio for the inductive skeleton fitting and showing it has smaller risk when the degree is less than three, with numerical verification and applications in generalized location problems and hyper-parameter tuning.
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 Bézier 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.