Achieving Sublinear Regret in Fixed-Size Sketching Linear Bandits: A Clustering Approach

We study the problem of computational efficiency for linear bandits in high-dimensional settings with a finite arm set. In linear bandits, the increase in the dimension $d$ of the feature vectors leads to growing computational costs of $O(d^2)$ at each round of update. Traditional sketching-based methods such as SOFUL reduce computation via fixed-size matrix sketching, yet run the risk of incurring vacuous linear regret when the spectral tail of the data is heavy and the sketch size is inadequately selected. To guarantee regret convergence and effectively reduce computational costs, we introduce a clustering mechanism and propose the Cluster Sketch Linear Bandit (CS-LB) algorithm. Our method preserves the full covariance information in each cluster to guarantee robust sublinear regret without spectral-tail vulnerabilities, performs cluster switching by assigning a sentinel for each cluster, and reduces per-round update computation to $O(l^{2}d)$ via a tunable sketch size $l<d$. Experiments on synthetic datasets demonstrate that our method consistently maintains a favorable trade-off between efficiency and regret.