梳理算子学习的收敛速率与统计极限，揭示学习性能边界。
A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
- 从经验风险最小化出发，分析全纯算子与神经网络逼近误差
- 通过极小极大视角，揭示样本量下的理论性能下界
- 探讨不同正则性假设下的学习限制，适合理论研究者
本文综述算子学习、统计学习理论与逼近论交叉领域的最新进展。首先回顾了基于经验风险最小化的误差界，重点关注全纯算子与神经网络近似情形。接着，采用极小极大视角，分析在多种超越全纯性的正则性假设下，样本量对学习性能的根本限制。最后讨论两种分析框架的相互关系，并提出若干开放问题。
原文摘要 · Abstract (English)
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.
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