Linear Ensemble Sampling with Fewer Ensembles

Ensemble sampling is a practically attractive approach to randomized exploration, but existing theoretical guarantees for linear bandits require ensembles much larger than what its practical motivation would suggest. In particular, the sharpest existing analysis achieves the ensemble complexity of $\Theta(d\log T)$, leaving a $\log T$ gap from the intrinsic $\Omega(d)$ ensemble-size barrier. We aim to narrow this gap by proposing an ensemble sampling algorithm that refreshes the ensemble only when the regularized Gram matrix changes substantially. This mechanism localizes the perturbation analysis to epochs with controlled Gram-matrix drift and reduces the sufficient ensemble size to $\Theta(d\log d+d\log\log T)$, while preserving the state-of-the-art $\tilde{\mathcal O}(d^{3/2}\sqrt T)$ regret for ensemble sampling with arbitrary bounded arm sets. We further show that, when the arm set is finite of cardinality $K$, the proposed algorithm achieves the sharper regret bound $\tilde{\mathcal{O}}(d\sqrt{T\log K})$. To the best of our knowledge, this is the first ensemble-sampling guarantee that simultaneously recovers both canonical regret scalings known for randomized linear bandit algorithms: the $\tilde{\mathcal{O}}(d^{3/2}\sqrt{T})$ rate for arbitrary bounded arm sets and the $\tilde{\mathcal{O}}(d\sqrt{T\log K})$ rate for finite arm sets. The algorithm also admits an anytime implementation without resetting past data, and experiments show that it remains competitive with baselines while using substantially smaller ensembles.