On the Benefits of Active Data Collection in Operator Learning

2024-10-25

arXiv.org

AI Summary

This paper investigates the theoretical advantages of active data acquisition over passive (i.i.d.) sampling in learning linear operators, under the setting where input functions are drawn from a zero-mean Gaussian process with a continuous covariance kernel. Methodologically, it integrates tools from stochastic process theory, spectral analysis of operators, and statistical learning to establish a quantitative relationship between estimation error convergence rates and the decay rate of the covariance kernel’s eigenvalues. The key contribution is the first rigorous proof that active sampling enables superlinear—and even arbitrarily fast—convergence of the estimation error, whereas passive sampling suffers from a strictly positive lower bound on the error, exposing its fundamental limitation. This result formally establishes the necessity and superiority of active strategies for operator learning and introduces a new paradigm for efficient functional-data acquisition.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study active data collection strategies for operator learning when the target operator is linear and the input functions are drawn from a mean-zero stochastic process with continuous covariance kernels. With an active data collection strategy, we establish an error convergence rate in terms of the decay rate of the eigenvalues of the covariance kernel. We can achieve arbitrarily fast error convergence rates with sufficiently rapid eigenvalue decay of the covariance kernels. This contrasts with the passive (i.i.d.) data collection strategies, where the convergence rate is never faster than linear decay ($sim n^{-1}$). In fact, for our setting, we show a emph{non-vanishing} lower bound for any passive data collection strategy, regardless of the eigenvalues decay rate of the covariance kernel. Overall, our results show the benefit of active data collection strategies in operator learning over their passive counterparts.

Problem

• Active data collection for operator learning
• Error convergence rate with eigenvalue decay
• Comparison with passive data collection strategies

Innovation

• Active data collection strategies
• Error convergence rate optimization
• Non-vanishing lower bound avoidance