# Local Function Complexity for Active Learning via Mixture of Gaussian Processes

## TL;DR

This paper introduces a Gaussian process-based local function complexity measure to enhance active learning, demonstrating improved efficiency in reconstructing quantum chemical force fields with limited data.

## Contribution

It develops a robust, scalable GPR-based local complexity measure and integrates it into an active learning framework for real-world, high-dimensional problems.

## Key findings

- Effective in low-dimensional synthetic data
- Achieves state-of-the-art performance in quantum chemistry task
- Reduces training data requirements significantly

## Abstract

Inhomogeneities in real-world data, e.g., due to changes in the observation noise level or variations in the structural complexity of the source function, pose a unique set of challenges for statistical inference. Accounting for them can greatly improve predictive power when physical resources or computation time is limited. In this paper, we draw on recent theoretical results on the estimation of local function complexity (LFC), derived from the domain of local polynomial smoothing (LPS), to establish a notion of local structural complexity, which is used to develop a model-agnostic active learning (AL) framework. Due to its reliance on pointwise estimates, the LPS model class is not robust and scalable concerning large input space dimensions that typically come along with real-world problems. Here, we derive and estimate the Gaussian process regression (GPR)-based analog of the LPS-based LFC and use it as a substitute in the above framework to make it robust and scalable. We assess the effectiveness of our LFC estimate in an AL application on a prototypical low-dimensional synthetic dataset, before taking on the challenging real-world task of reconstructing a quantum chemical force field for a small organic molecule and demonstrating state-of-the-art performance with a significantly reduced training demand.