mlsanityData to 14 Oct 2026

Sequential operator learning under dependent data

arXiv 2608.24426 ↗First posted 25 Aug 2026stat.ML cs.LGcited by 0

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Abstract (arXiv)

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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References · 20

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  3. 2017On Kernelized Multi-armed Banditscited by 352
  4. 2022FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operatorscited by 188
  5. 2020On Information Gain and Regret Bounds in Gaussian Process Banditscited by 158
  6. 2023A Mathematical Guide to Operator Learningcited by 76
  7. 2020Learning nonlinear dynamical systems from a single trajectorycited by 46
  8. 2023On the training and generalization of deep operator networkscited by 31
  9. 2023Multi-Resolution Active Learning of Fourier Neural Operatorscited by 22
  10. 2024Active Learning for Neural PDE Solverscited by 22
  11. 2020No-regret Algorithms for Multi-task Bayesian Optimizationcited by 18
  12. 2024Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networkscited by 10
8 more
  1. 2020Value Function Approximations via Kernel Embeddings for No-Regret Reinforcement Learningcited by 7
  2. 2024On the Benefits of Active Data Collection in Operator Learningcited by 5
  3. 2025Vector-valued self-normalized concentration inequalities beyond sub-Gaussianitycited by 4
  4. 2025A variational approach to dimension-free self-normalized concentrationcited by 2
  5. 2025Thompson Sampling in Function Spaces via Neural Operatorscited by 1
  6. 2026Multi-Scale Wavelet Transformers for Operator Learning of Dynamical Systemscited by 1
  7. 2026Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Frameworkcited by 1
  8. 2026Kernel-based guarantees for nonlinear parametric models in Bayesian optimizationcited by 1

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