Sequential operator learning under dependent data

2026-08-25

AI Summary

研究解决了从依赖数据中学习操作符的问题，通过推导希尔伯特空间中随机过程的时间均匀自归一化集中界，为线性和非线性操作符提供了回归误差保证。

Abstract

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.

Problem

• sequential operator learning
• dependent data
• adaptive experimental design
• Bayesian optimization
• dynamical-system modelling

Innovation

• time-uniform self-normalized concentration bounds
• Hilbert spaces
• adaptive operator learning
• stochastic dynamical data
• strongly convex losses