It is well-known that the worst-case minimax regret for sparse linear bandits is $\widetilde{\Theta}\left(\sqrt{dT}\right)$ where $d$ is the ambient dimension and $T$ is the number of time steps (ignoring the dependency on sparsity). On the other hand, in the benign setting where there is no noise and the action set is the unit sphere, one can use divide-and-conquer to achieve an $\widetilde{\mathcal O}(1)$ regret, which is (nearly) independent of $d$ and $T$. In this paper, we present the first variance-aware regret guarantee for sparse linear bandits: $\widetilde{\mathcal O}\left(\sqrt{d\sum_{t=1}^T \sigma_t^2} + 1\right)$, where $\sigma_t^2$ is the variance of the noise at the $t$-th time step. This bound naturally interpolates the regret bounds for the worst-case constant-variance regime ($\sigma_t = \Omega(1)$) and the benign deterministic regimes ($\sigma_t = 0$). To achieve this variance-aware regret guarantee, we develop a general framework that converts any variance-aware linear bandit algorithm to a variance-aware algorithm for sparse linear bandits in a ``black-box'' manner. Specifically, we take two recent algorithms as black boxes to illustrate that the claimed bounds indeed hold, where the first algorithm can handle unknown-variance cases and the second one is more efficient.

本文发展了一种数据稀疏(稀疏线性斯托剖斯)的新型算法，该算法使较小的环境维度下研究致命后果的风险得以降低，实现了一种有区别的、具有人造和样本方差适应性的总体安全算法。同时，该研究还探讨了如何在一种<黑盒>化的方式下将任何方差算法转化为sparce linear bandits算法。

方差感知稀疏线性赌博机