A short tour of operator learning theory: Convergence rates, statistical limits, and open questions

2026-02-28

AI Summary

This study investigates the theoretical foundations of operator learning, with a focus on convergence rates and fundamental statistical limits. By integrating tools from statistical learning theory, approximation theory, and the framework of holomorphic operators, the work establishes a unified error analysis framework to systematically derive generalization error bounds for empirical risk minimization. Under generalized regularity conditions, it further establishes minimax-optimal statistical lower bounds, revealing an intrinsic trade-off between sample complexity and model approximation capacity. The analysis delineates current theoretical boundaries in operator learning and identifies several key open problems, offering new perspectives to guide future theoretical advances in the field.

Technology Category

• Machine Learning: Learning Theory
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.

Problem

• operator learning
• convergence rates
• statistical limits
• minimax theory
• regularity

Innovation

• operator learning
• empirical risk minimization
• minimax rates
• holomorphic operators
• statistical limits