Approximate Thompson sampling with Langevin Monte Carlo broadens its reach from Gaussian posterior sampling to encompass more general smooth posteriors. However, it still encounters scalability issues in high-dimensional problems when demanding high accuracy. To address this, we propose an approximate Thompson sampling strategy, utilizing underdamped Langevin Monte Carlo, where the latter is the go-to workhorse for simulations of high-dimensional posteriors. Based on the standard smoothness and log-concavity conditions, we study the accelerated posterior concentration and sampling using a specific potential function. This design improves the sample complexity for realizing logarithmic regrets from $\mathcal{\tilde O}(d)$ to $\mathcal{\tilde O}(\sqrt{d})$. The scalability and robustness of our algorithm are also empirically validated through synthetic experiments in high-dimensional bandit problems.

翻译：近似汤普森采样结合朗之万蒙特卡洛方法，将其适用范围从高斯后验采样扩展至更一般的平滑后验分布。然而，在高维问题中追求高精度时仍存在可扩展性难题。为此，我们提出了一种基于欠阻尼朗之万蒙特卡洛的近似汤普森采样策略 - 后者正是高维后验模拟的核心工具。基于标准光滑性与对数凹性条件，我们通过特定势能函数研究了加速后验浓度与采样过程。该设计将对数遗憾的样本复杂度从$\mathcal{\tilde O}(d)$提升至$\mathcal{\tilde O}(\sqrt{d})$。通过高维赌博机问题的合成实验，我们实证验证了算法的可扩展性与鲁棒性。