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.

翻译：水平集估计问题旨在识别候选点集内，某个未知且评估成本高昂的函数值超过指定阈值的区域，为穷举评估函数值提供了一种高效替代方案。传统方法通常采用序贯优化策略寻找$ε$-精确解，此类解允许在阈值等高线附近留有余量，但往往缺乏有效的停机准则，导致过度探索和低效。本文提出了一种融合停机准则的水平集估计采集策略，确保算法在进一步探索无法带来改进时停止，从而减少不必要的函数评估。我们从理论上证明，该方法能以$1 - δ$的置信水平满足$ε$-精度要求，弥补了现有方法的一个关键空白。此外，我们展示了这一点还能保证F-score等性能指标的下界。数值实验表明，所提出的采集函数在达到与现有方法相当的精确度的同时，证实了停机准则能在充分探索后有效终止算法。