Active Learning for Level Set Estimation Using Randomized Straddle Algorithms

TL;DR

This paper introduces a randomized straddle algorithm for level set estimation that replaces the confidence parameter with a chi-squared distribution sample, eliminating the need for parameter tuning and providing theoretical guarantees.

Contribution

The paper proposes a novel randomized approach to the straddle algorithm that simplifies parameter selection and guarantees performance based on sample complexity.

Findings

The randomized method performs well on synthetic data.

It outperforms traditional methods with conservative confidence parameters.

Theoretical guarantees depend on sample complexity and iterations.

Abstract

Level set estimation (LSE), the problem of identifying the set of input points where a function takes value above (or below) a given threshold, is important in practical applications. When the function is expensive-to-evaluate and black-box, the \textit{straddle} algorithm, which is a representative heuristic for LSE based on Gaussian process models, and its extensions having theoretical guarantees have been developed. However, many of existing methods include a confidence parameter $\beta^{1/2}_t$ that must be specified by the user, and methods that choose $\beta^{1/2}_t$ heuristically do not provide theoretical guarantees. In contrast, theoretically guaranteed values of $\beta^{1/2}_t$ need to be increased depending on the number of iterations and candidate points, and are conservative and not good for practical performance. In this study, we propose a novel method, the \textit{randomized straddle} algorithm, in which $\beta_t$ in the straddle algorithm is replaced by a random sample from the chi-squared distribution with two degrees of freedom. The confidence parameter in the proposed method has the advantages of not needing adjustment, not depending on the number of iterations and candidate points, and not being conservative. Furthermore, we show that the proposed method has theoretical guarantees that depend on the sample complexity and the number of iterations. Finally, we confirm the usefulness of the proposed method through numerical experiments using synthetic and real data.