It is well-known that for sparse linear bandits, when ignoring the dependency on sparsity which is much smaller than the ambient dimension, the worst-case minimax regret is $\widetilde{\Theta}\left(\sqrt{dT}\right)$ where $d$ is the ambient dimension and $T$ is the number of rounds. 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 $\widetilde{\mathcal O}(1)$ 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 round. This bound naturally interpolates the regret bounds for the worst-case constant-variance regime (i.e., $\sigma_t \equiv \Omega(1)$) and the benign deterministic regimes (i.e., $\sigma_t \equiv 0$). 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.

翻译：众所周知，对于稀疏线性赌博机，当忽略远小于环境维度的稀疏性依赖性时，最坏情况下的极小极大遗憾值为$\widetilde{\Theta}\left(\sqrt{dT}\right)$，其中$d$为环境维度，$T$为轮数。另一方面，在无噪声且动作集为单位球面的良性设定下，可采用分治策略实现$\widetilde{\mathcal O}(1)$的遗憾值，该结果（几乎）与$d$和$T$无关。本文首次提出稀疏线性赌博机的方差感知遗憾保证：$\widetilde{\mathcal O}\left(\sqrt{d\sum_{t=1}^T \sigma_t^2} + 1\right)$，其中$\sigma_t^2$表示第$t$轮噪声的方差。该界限自然地插值了最坏情况下的恒定方差设定（即$\sigma_t \equiv \Omega(1)$）与良性确定性设定（即$\sigma_t \equiv 0$）之间的遗憾界。为获得此方差感知遗憾保证，我们开发了一个通用框架，能以"黑盒"方式将任意方差感知线性赌博机算法转换为稀疏线性赌博机的方差感知算法。具体而言，我们以两种近期算法作为黑盒来验证所声称界限的成立性——其中第一种算法能处理未知方差情形，第二种算法则具有更高效率。