发表机构
CSIRO Technology（联邦科学与工业研究组织科技部门）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对依赖数据下的序列算子学习问题，推导希尔伯特空间中随机过程的时间一致自正则集中界，为线性与非线性参数算子提供回归误差保证，为自适应算子学习等场景的收敛保证提供重要进展。
AI 中文摘要
从序列采集的数据中学习算子的场景出现在自适应实验设计、贝叶斯优化与动力系统建模中，这类场景下的观测值可能存在依赖关系，且未来的输入或感知算子可能依赖于先前的数据。我们推导了带向量值噪声的希尔伯特空间中随机过程的时间一致自正则集中界，利用这些界获得线性算子（包括希尔伯特估计空间外的目标）及采用强凸损失与正则化项训练的非线性参数算子的回归误差保证。我们的结果允许输入与输出为无穷维，无需独立性或混合性假设，为自适应算子学习与从随机动力数据中学习的收敛保证迈出了重要一步。
英文摘要
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.
CommentsUnder review. Work in progress
