@inproceedings{
ito2024on,
title={On the Minimax Regret for Contextual Linear Bandits and Multi-Armed Bandits with Expert Advice},
author={Shinji Ito},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=AkiPax5SXu}
}
This paper examines two extensions of multi-armed bandit problems: multi-armed bandits with expert advice and contextual linear bandits.  For the former problem, multi-armed bandits with expert advice, the previously known best upper and lower bounds have been $O(\sqrt{KT \log \frac{N}{K} })$ and $\Omega( \sqrt{KT \frac{ \log N }{\log K }} )$, respectively.  Here, $K$, $N$, and $T$ represent the numbers of arms, experts, and rounds, respectively.  This paper closes the gap between these bounds by presenting a matching lower bound of $\Omega( \sqrt{KT \log \frac{N}{K}} )$. 
  This lower bound is shown for the problem setting in which the player chooses an expert before observing the advices in each round.
  For the latter problem, contextual linear bandits, we provide an algorithm that achieves $O ( \sqrt{d T \log ( K \min\{ 1, \frac{S}{d} \} )} )$ together with a matching lower bound, where $d$ and $S$ represent the dimensionality of feature vectors and the size of the context space, respectively.
On the Minimax Regret for Contextual Linear Bandits and Multi-Armed Bandits with Expert Advice
Shinji Ito