提出依赖数据下算子学习的统一理论，为自适应实验设计等场景提供严格误差保证。
Sequential operator learning under dependent data
- 基于希尔伯特空间中的自归一化不等式，处理带向量噪声的随机过程
- 在无独立性假设下，给出线性与非线性算子的回归误差上界
- 适用于无限维输入输出，适合动态系统建模与贝叶斯优化研究者
从序列采集的数据中学习算子出现在自适应实验设计、贝叶斯优化和动力系统建模中，其中观测值可能相关，且未来的输入或传感算子可能依赖于先前数据。本文推导了希尔伯特空间中具有向量值噪声的随机过程的时间一致自归一化浓度不等式。利用这些不等式，获得了线性算子（包括估计空间外的目标）以及用强凸损失和正则项训练的非线性参数化算子的回归误差保证。结果无需独立性或混合性假设，支持可能无限维的输入与输出，为自适应算子学习和随机动力系统数据学习的收敛性分析提供了重要进展。
原文摘要 · Abstract (English)
Learning operators from sequentially collected data arises in adaptive experimental design, Bayesian optimization, and dynamical-system modelling, where observations may be dependent, and future inputs or sensing operators may depend on preceding data. We derive time-uniform self-normalized concentration bounds for stochastic processes in Hilbert spaces with vector-valued noise. We use these bounds to obtain regression-error guarantees for linear operators, including targets outside the Hilbert estimation space, and for nonlinear parametric operators trained with strongly convex losses and regularizers. Our results allow possibly infinite-dimensional inputs and outputs without independence or mixing assumptions, providing a major step towards convergence guarantees for adaptive operator learning and learning from stochastic dynamical data.
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