# Streaming kernel regression with provably adaptive mean, variance, and regularization

arXiv: https://arxiv.org/abs/1708.00768 · HTML: https://mlsanity.com/paper/1708.00768/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2017.

## Built on by (most cited first, 15 of 15)

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- [Scaling Gaussian Process Optimization by Evaluating a Few Unique Candidates Multiple Times](https://mlsanity.com/paper/2201.12909/) (2022)
- [Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures](https://mlsanity.com/paper/2309.14298/) (2023)
- [A Robust Phased Elimination Algorithm for Corruption-Tolerant Gaussian Process Bandits](https://mlsanity.com/paper/2202.01850/) (2022)
- [Batch Bayesian optimisation via density-ratio estimation with guarantees](https://mlsanity.com/paper/2209.10715/) (2022)
- [Tighter Confidence Bounds for Sequential Kernel Regression](https://mlsanity.com/paper/2403.12732/) (2024)
- [Variational Search Distributions](https://mlsanity.com/paper/2409.06142/) (2024)
- [Indexed Minimum Empirical Divergence for Unimodal Bandits](https://mlsanity.com/paper/2112.01452/) (2021)
- [Generative Bayesian Optimization: Generative Models as Acquisition Functions](https://mlsanity.com/paper/2510.25240/) (2025)
- [Kernel-based guarantees for nonlinear parametric models in Bayesian optimization](https://mlsanity.com/paper/2605.13160/) (2026)
- [Bandits for Efficient Experimentation: Adapting to Control Group, Preferences, and Context Drifts](https://mlsanity.com/paper/2606.09802/) (2026)

## Abstract (arXiv)

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bersntein-like concentration bounds. The later is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.