Asymptotic Risk of Bézier Simplex Fitting

Abstract

The B'ezier 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-at-once fitting uses a non-stratified sampling which treats a simplex as a single object. 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 B'ezier 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 B'ezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a B'ezier simplex of degree three where the all-at-once fitting gets an advantage.

Topology of Pareto sets of strongly convex problems

“…( 11) is weakly simplicial, the mapping θ * : ∆ 1 → R p in eq. ( 12) can be approximated by a Bézier simplex with small samples (see [24,Section 4]).…”

“…The property that Φ sends the faces to the Pareto sets has a great utility: it is known that using this correspondence, we can reduce the number of solutions to obtain an approximation of the mapping Φ, i.e., a parametric-surface approximation of the Pareto set (cf. [12,24]).…”

“…Since the scalarized function m i=1 w i f i : R n → R is strongly convex (see section 2.3), the mapping x * is well-defined and can be efficiently computed by, e.g., gradient methods. By this property, scalarization-based subdivision methods [3,6,17,23] are able to provide solutions for parametric-surface approximation methods in [12,24] to build an approximation of Φ. 1 While it has been stated that any convex problem is simplicial in the literature, it is too optimistic to expect so, as one can easily find a counterexample.…”

Characterization of the equality of weak efficiency and efficiency on convex free disposal hulls

“…While we cannot express (7.5) in a closed form, the mapping x * : ∆ m−1 → S(f, R n ) can be approximated by a Bézier simplex [11,Theorem 3.1]. For numerical computation, see [27].…”

“…f • θ * ) for this dataset seems to be continuous. Motivated by the above observation, the authors applied a Bézier simplex fitting method [27] to this sample and obtained an approximation of θ * and f • θ * (Figure 7). As shown in the figure, the mappings θ * and f • θ * are accurately approximated.…”