The utilization of sketching techniques has progressively emerged as a pivotal method for enhancing the efficiency of online learning. In linear bandit settings, current sketch-based approaches leverage matrix sketching to reduce the per-round time complexity from \(\Omega\left(d^2\right)\) to \(O(d)\), where \(d\) is the input dimension. Despite this improved efficiency, these approaches encounter critical pitfalls: if the spectral tail of the covariance matrix does not decrease rapidly, it can lead to linear regret. In this paper, we revisit the regret analysis and algorithm design concerning approximating the covariance matrix using matrix sketching in linear bandits. We illustrate how inappropriate sketch sizes can result in unbounded spectral loss, thereby causing linear regret. To prevent this issue, we propose Dyadic Block Sketching, an innovative streaming matrix sketching approach that adaptively manages sketch size to constrain global spectral loss. This approach effectively tracks the best rank-\( k \) approximation in an online manner, ensuring efficiency when the geometry of the covariance matrix is favorable. Then, we apply the proposed Dyadic Block Sketching to linear bandits and demonstrate that the resulting bandit algorithm can achieve sublinear regret without prior knowledge of the covariance matrix, even under the worst case. Our method is a general framework for efficient sketch-based linear bandits, applicable to all existing sketch-based approaches, and offers improved regret bounds accordingly. Additionally, we conduct comprehensive empirical studies using both synthetic and real-world data to validate the accuracy of our theoretical findings and to highlight the effectiveness of our algorithm.

翻译：素描技术的应用已逐渐成为提升在线学习效率的关键方法。在线性赌博机场景中，当前基于素描的方法利用矩阵素描技术将每轮时间复杂度从\(\Omega\left(d^2\right)\)降低至\(O(d)\)，其中\(d\)为输入维度。尽管效率得到提升，这些方法仍面临关键缺陷：若协方差矩阵的谱尾端未快速衰减，可能导致线性遗憾。本文重新审视了在线性赌博机中使用矩阵素描近似协方差矩阵的遗憾分析与算法设计。我们阐明了不恰当的素描尺寸如何导致无界谱损失，从而引发线性遗憾。为解决此问题，我们提出了Dyadic Block Sketching——一种创新的流式矩阵素描方法，能自适应管理素描尺寸以约束全局谱损失。该方法能有效在线追踪最优秩-\(k\)近似，在协方差矩阵几何特性良好时确保效率。随后，我们将所提出的Dyadic Block Sketching应用于线性赌博机，证明所得赌博机算法即使在最坏情况下，无需先验协方差矩阵知识即可实现次线性遗憾。我们的方法是基于素描的高效线性赌博机通用框架，适用于所有现有素描方法，并能相应改进遗憾界。此外，我们通过合成数据与真实数据进行了全面实证研究，验证理论发现的准确性并突显算法的有效性。