  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.
