Sequential operator learning under dependent data
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.
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BibTeX
@misc{oliveira2026sequential,
title = {{Sequential operator learning under dependent data}},
author = {Rafael Oliveira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24426},
eprint = {2608.24426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24426},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sequential operator learning under dependent data
AU - Rafael Oliveira
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.24426
UR - https://arxiv.org/abs/2608.24426
AB - 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.
ER -