# Active learning for enumerating local minima based on Gaussian process derivatives

## TL;DR

This paper introduces an active learning approach using Gaussian Processes to efficiently identify all local minima of a black-box function by sequentially selecting points based on derivative confidence intervals.

## Contribution

The paper proposes a novel active learning method that leverages GP derivatives to enumerate local minima, with theoretical analysis and numerical validation.

## Key findings

- Effective enumeration of local minima achieved
- The method outperforms baseline approaches in experiments
- Theoretical guarantees support the approach's validity

## Abstract

We study active learning (AL) based on Gaussian Processes (GPs) for efficiently enumerating all of the local minimum solutions of a black-box function. This problem is challenging due to the fact that local solutions are characterized by their zero gradient and positive-definite Hessian properties, but those derivatives cannot be directly observed. We propose a new AL method in which the input points are sequentially selected such that the confidence intervals of the GP derivatives are effectively updated for enumerating local minimum solutions. We theoretically analyze the proposed method and demonstrate its usefulness through numerical experiments.