Stochastic Linear Bandits with Parameter Noise

Daniel Ezer, Alon Cohen, Yishay Mansour
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:28463-28491, 2026.

Abstract

We study the stochastic linear bandits with parameter noise model, in which the reward of action $a$ is $a^\top \theta$ where $\theta$ is sampled i.i.d. We show a regret upper bound of $\widetilde{O} (\sqrt{d T \log(K/\delta) \sigma^2_{\max}})$ for a horizon $T$, general action set of size $K$ of dimension $d$, and where $\sigma^2_{\max}$ is the maximal variance of the reward for any action. We further provide a lower bound of $\widetilde{\Omega} (d \sqrt{T \sigma_{\max}^2})$ which is tight (up to logarithmic factors) whenever $\log K \approx d$. For more specific action sets, $\ell_p$ unit balls with $p \leq 2$ and dual norm $q$, we show that the minimax regret is $\widetilde{\Theta} (\sqrt{dT \sigma_q^2})$, where $\sigma_q^2$ is a variance-dependent quantity that is always at most $4$. This is in contrast to the minimax regret attainable for such sets in the classic additive noise model where the regret is of order $d \sqrt{T}$. Surprisingly, we show that this optimal (up to logarithmic factors) regret bound is attainable using a very simple explore-exploit algorithm.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-ezer26a, title = {Stochastic Linear Bandits with Parameter Noise}, author = {Ezer, Daniel and Cohen, Alon and Mansour, Yishay}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {28463--28491}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/ezer26a/ezer26a.pdf}, url = {https://proceedings.mlr.press/v306/ezer26a.html}, abstract = {We study the stochastic linear bandits with parameter noise model, in which the reward of action $a$ is $a^\top \theta$ where $\theta$ is sampled i.i.d. We show a regret upper bound of $\widetilde{O} (\sqrt{d T \log(K/\delta) \sigma^2_{\max}})$ for a horizon $T$, general action set of size $K$ of dimension $d$, and where $\sigma^2_{\max}$ is the maximal variance of the reward for any action. We further provide a lower bound of $\widetilde{\Omega} (d \sqrt{T \sigma_{\max}^2})$ which is tight (up to logarithmic factors) whenever $\log K \approx d$. For more specific action sets, $\ell_p$ unit balls with $p \leq 2$ and dual norm $q$, we show that the minimax regret is $\widetilde{\Theta} (\sqrt{dT \sigma_q^2})$, where $\sigma_q^2$ is a variance-dependent quantity that is always at most $4$. This is in contrast to the minimax regret attainable for such sets in the classic additive noise model where the regret is of order $d \sqrt{T}$. Surprisingly, we show that this optimal (up to logarithmic factors) regret bound is attainable using a very simple explore-exploit algorithm.} }
Endnote
%0 Conference Paper %T Stochastic Linear Bandits with Parameter Noise %A Daniel Ezer %A Alon Cohen %A Yishay Mansour %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-ezer26a %I PMLR %P 28463--28491 %U https://proceedings.mlr.press/v306/ezer26a.html %V 306 %X We study the stochastic linear bandits with parameter noise model, in which the reward of action $a$ is $a^\top \theta$ where $\theta$ is sampled i.i.d. We show a regret upper bound of $\widetilde{O} (\sqrt{d T \log(K/\delta) \sigma^2_{\max}})$ for a horizon $T$, general action set of size $K$ of dimension $d$, and where $\sigma^2_{\max}$ is the maximal variance of the reward for any action. We further provide a lower bound of $\widetilde{\Omega} (d \sqrt{T \sigma_{\max}^2})$ which is tight (up to logarithmic factors) whenever $\log K \approx d$. For more specific action sets, $\ell_p$ unit balls with $p \leq 2$ and dual norm $q$, we show that the minimax regret is $\widetilde{\Theta} (\sqrt{dT \sigma_q^2})$, where $\sigma_q^2$ is a variance-dependent quantity that is always at most $4$. This is in contrast to the minimax regret attainable for such sets in the classic additive noise model where the regret is of order $d \sqrt{T}$. Surprisingly, we show that this optimal (up to logarithmic factors) regret bound is attainable using a very simple explore-exploit algorithm.
APA
Ezer, D., Cohen, A. & Mansour, Y.. (2026). Stochastic Linear Bandits with Parameter Noise. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:28463-28491 Available from https://proceedings.mlr.press/v306/ezer26a.html.

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