  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.
