An $(ε,δ)$-accurate level set estimation with a stopping criterion
Hideaki Ishibashi, Kota Matsui, Kentaro Kutsukake, Hideitsu Hino
TL;DR
This work addresses level set estimation for costly black-box functions by introducing an acquisition function that directly targets classification difficulty and a stopping criterion with probabilistic guarantees. By modeling the unknown function with a Gaussian process, the method computes misclassification probabilities and a margin-based undetermined region to drive queries, while the stopping rule ensures the estimated level-set triplet is $\\epsilon$-accurate with probability at least $1-\\delta$. It further provides lower bounds for performance metrics such as the F-score and demonstrates competitiveness relative to established methods on synthetic benchmarks and a real-world red-zone estimation task in silicon ingots. The approach enables substantial reductions in function evaluations through principled early stopping, with practical implications for adaptive experimental design in materials quality and related domains.
Abstract
The level set estimation problem seeks to identify regions within a set of candidate points where an unknown and costly to evaluate function's value exceeds a specified threshold, providing an efficient alternative to exhaustive evaluations of function values. Traditional methods often use sequential optimization strategies to find $ε$-accurate solutions, which permit a margin around the threshold contour but frequently lack effective stopping criteria, leading to excessive exploration and inefficiencies. This paper introduces an acquisition strategy for level set estimation that incorporates a stopping criterion, ensuring the algorithm halts when further exploration is unlikely to yield improvements, thereby reducing unnecessary function evaluations. We theoretically prove that our method satisfies $ε$-accuracy with a confidence level of $1 - δ$, addressing a key gap in existing approaches. Furthermore, we show that this also leads to guarantees on the lower bounds of performance metrics such as F-score. Numerical experiments demonstrate that the proposed acquisition function achieves comparable precision to existing methods while confirming that the stopping criterion effectively terminates the algorithm once adequate exploration is completed.