Asymptotic Risk of Bézier Simplex Fitting

The Bézier simplex fitting is a novel data modeling technique which utilizes geometric structures of data to approximate the Pareto set of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from skeletons of a simplex, whereas the all-atonce fitting uses a non-stratified sampling which treats a simplex as a single object. 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 Bézier 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 Bézier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a Bézier simplex of degree three where the all-at-once fitting gets an advantage.

Authors

• Akinori Tanaka
• Akiyoshi Sannai
• Ken Kobayashi
• Naoki Hamada KLab Inc.

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