Linear bandits have become a cornerstone of online learning and sequential decision-making, providing solid theoretical foundations for balancing exploration and exploitation. Within this domain, matrix sketching serves as a critical component for achieving computational efficiency, especially when confronting high-dimensional problem instances. The sketch-based approaches reduce per-round complexity from $Ω(d^2)$ to $O(dl)$, where $d$ is the dimension and $l<d$ is the sketch size. However, this computational efficiency comes with a fundamental pitfall: when the streaming matrix exhibits heavy spectral tails, such algorithms can incur vacuous \textit{linear regret}. In this paper, we revisit the regret bounds and algorithmic design for sketch-based linear bandits. Our analysis reveals that inappropriate sketch sizes can lead to substantial spectral error, severely undermining regret guarantees. To overcome this issue, we propose Dyadic Block Sketching, a novel multi-scale matrix sketching approach that dynamically adjusts the sketch size during the learning process. We apply this technique to linear bandits and demonstrate that the new algorithm achieves \textit{sublinear regret} bounds without requiring prior knowledge of the streaming matrix properties. It establishes a general framework for efficient sketch-based linear bandits, which can be integrated with any matrix sketching method that provides covariance guarantees. Comprehensive experimental evaluation demonstrates the superior utility-efficiency trade-off achieved by our approach.

翻译：线性赌博机已成为在线学习和序贯决策的基石，为探索与利用的平衡提供了坚实的理论基础。在该领域中，矩阵素描是实现计算效率的关键组成部分，尤其是在面对高维问题实例时。基于素描的方法将每轮复杂度从 $Ω(d^2)$ 降低至 $O(dl)$，其中 $d$ 为维度，$l<d$ 为素描尺寸。然而，这种计算效率伴随着一个根本性缺陷：当流矩阵呈现厚重的谱尾时，此类算法可能招致空洞的\textit{线性遗憾}。本文重新审视了基于素描的线性赌博机的遗憾界与算法设计。我们的分析表明，不恰当的素描尺寸会导致显著的谱误差，从而严重削弱遗憾保证。为克服此问题，我们提出了二元块素描，一种新颖的多尺度矩阵素描方法，可在学习过程中动态调整素描尺寸。我们将此技术应用于线性赌博机，并证明新算法实现了\textit{次线性遗憾}界，且无需预先知晓流矩阵的特性。这为高效的基于素描的线性赌博机建立了一个通用框架，该框架可与任何提供协方差保证的矩阵素描方法集成。全面的实验评估表明，我们的方法实现了优越的效用-效率权衡。