Linear Bandits on Ellipsoids: Minimax Optimal Algorithms

Raymond Zhang, Hadiji Hédi, Combes Richard
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:6016-6040, 2025.

Abstract

We consider linear stochastic bandits where the set of actions is an ellipsoid. We provide the first known minimax optimal algorithm for this problem. We first derive a novel information-theoretic lower bound on the regret of any algorithm, which must be at least $\Omega(\min(d \sigma \sqrt{T} + d \|\theta\|_{A}, \|\theta\|_{A} T))$ where $d$ is the dimension, $T$ the time horizon, $\sigma^2$ the noise variance, $A$ a matrix defining the set of actions and $\theta$ the vector of unknown parameters. We then provide an algorithm whose regret matches this bound to a multiplicative universal constant. The algorithm is non-classical in the sense that it is not optimistic, and it is not a sampling algorithm. The main idea is to combine a novel sequential procedure to estimate $\|\theta\|$, followed by an explore-and-commit strategy informed by this estimate. The algorithm is highly computationally efficient, and a run requires only time $O(dT + d^2 \log(T/d) + d^3)$ and memory $O(d^2)$, in contrast with known optimistic algorithms, which are not implementable in polynomial time. We go beyond minimax optimality and show that our algorithm is locally asymptotically minimax optimal, a much stronger notion of optimality. We further provide numerical experiments to illustrate our theoretical findings. The code to reproduce the experiments is available at \url{https://github.com/RaymZhang/LinearBanditsEllipsoidsMinimaxCOLT}.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-zhang25b, title = {Linear Bandits on Ellipsoids: Minimax Optimal Algorithms}, author = {Zhang, Raymond and H{\'e}di, Hadiji and Richard, Combes}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {6016--6040}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/zhang25b/zhang25b.pdf}, url = {https://proceedings.mlr.press/v291/zhang25b.html}, abstract = {We consider linear stochastic bandits where the set of actions is an ellipsoid. We provide the first known minimax optimal algorithm for this problem. We first derive a novel information-theoretic lower bound on the regret of any algorithm, which must be at least $\Omega(\min(d \sigma \sqrt{T} + d \|\theta\|_{A}, \|\theta\|_{A} T))$ where $d$ is the dimension, $T$ the time horizon, $\sigma^2$ the noise variance, $A$ a matrix defining the set of actions and $\theta$ the vector of unknown parameters. We then provide an algorithm whose regret matches this bound to a multiplicative universal constant. The algorithm is non-classical in the sense that it is not optimistic, and it is not a sampling algorithm. The main idea is to combine a novel sequential procedure to estimate $\|\theta\|$, followed by an explore-and-commit strategy informed by this estimate. The algorithm is highly computationally efficient, and a run requires only time $O(dT + d^2 \log(T/d) + d^3)$ and memory $O(d^2)$, in contrast with known optimistic algorithms, which are not implementable in polynomial time. We go beyond minimax optimality and show that our algorithm is locally asymptotically minimax optimal, a much stronger notion of optimality. We further provide numerical experiments to illustrate our theoretical findings. The code to reproduce the experiments is available at \url{https://github.com/RaymZhang/LinearBanditsEllipsoidsMinimaxCOLT}. } }
Endnote
%0 Conference Paper %T Linear Bandits on Ellipsoids: Minimax Optimal Algorithms %A Raymond Zhang %A Hadiji Hédi %A Combes Richard %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-zhang25b %I PMLR %P 6016--6040 %U https://proceedings.mlr.press/v291/zhang25b.html %V 291 %X We consider linear stochastic bandits where the set of actions is an ellipsoid. We provide the first known minimax optimal algorithm for this problem. We first derive a novel information-theoretic lower bound on the regret of any algorithm, which must be at least $\Omega(\min(d \sigma \sqrt{T} + d \|\theta\|_{A}, \|\theta\|_{A} T))$ where $d$ is the dimension, $T$ the time horizon, $\sigma^2$ the noise variance, $A$ a matrix defining the set of actions and $\theta$ the vector of unknown parameters. We then provide an algorithm whose regret matches this bound to a multiplicative universal constant. The algorithm is non-classical in the sense that it is not optimistic, and it is not a sampling algorithm. The main idea is to combine a novel sequential procedure to estimate $\|\theta\|$, followed by an explore-and-commit strategy informed by this estimate. The algorithm is highly computationally efficient, and a run requires only time $O(dT + d^2 \log(T/d) + d^3)$ and memory $O(d^2)$, in contrast with known optimistic algorithms, which are not implementable in polynomial time. We go beyond minimax optimality and show that our algorithm is locally asymptotically minimax optimal, a much stronger notion of optimality. We further provide numerical experiments to illustrate our theoretical findings. The code to reproduce the experiments is available at \url{https://github.com/RaymZhang/LinearBanditsEllipsoidsMinimaxCOLT}.
APA
Zhang, R., Hédi, H. & Richard, C.. (2025). Linear Bandits on Ellipsoids: Minimax Optimal Algorithms. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:6016-6040 Available from https://proceedings.mlr.press/v291/zhang25b.html.

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