Sequential operator learning under dependent data

Learning operators from sequentially collected data arises in adaptive experimental design, Bayesian optimization, and dynamical-system modeling, 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 losses strongly convex in the predicted observations, but not necessarily in the parameters. Our results allow possibly infinite-dimensional inputs and outputs without independence or mixing assumptions, providing a general framework for analyzing adaptive operator learning and learning from stochastic dynamical data.