# Sequential operator learning under dependent data

arXiv: https://arxiv.org/abs/2608.24426 · HTML: https://mlsanity.com/paper/2608.24426/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 45 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [No-regret Algorithms for Multi-task Bayesian Optimization](https://mlsanity.com/paper/2008.08885/) (2020)
- [Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networks](https://mlsanity.com/paper/2404.03099/) (2024)
- [Value Function Approximations via Kernel Embeddings for No-Regret Reinforcement Learning](https://mlsanity.com/paper/2011.07881/) (2020)
- [On the Benefits of Active Data Collection in Operator Learning](https://mlsanity.com/paper/2410.19725/) (2024)
- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [A variational approach to dimension-free self-normalized concentration](https://mlsanity.com/paper/2508.06483/) (2025)
- [Thompson Sampling in Function Spaces via Neural Operators](https://mlsanity.com/paper/2506.21894/) (2025)
- [Multi-Scale Wavelet Transformers for Operator Learning of Dynamical Systems](https://mlsanity.com/paper/2602.01486/) (2026)
- [Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Framework](https://mlsanity.com/paper/2605.09775/) (2026)
- [Kernel-based guarantees for nonlinear parametric models in Bayesian optimization](https://mlsanity.com/paper/2605.13160/) (2026)

## 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.