Sequential operator learning under dependent data

• URL: http://arxiv.org/abs/2608.24426v1
• Date: Tue, 25 Aug 2026 11:42:09 GMT
• Title: Sequential operator learning under dependent data
• Abstract summary: Learning operators from sequentially collected data arise in adaptive experimental design, Bayesian optimization, and dynamical-system modelling.<n>We derive time-uniform self-normalized concentration bounds for processes in Hilbert spaces with vector-valued noise.<n>Our results allow possibly infinite-dimensional inputs and outputs without independence or mixing assumptions.
• Score: 5.191484562883949
• License: http://creativecommons.org/licenses/by/4.0/
• 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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