Abstract: In this work, we derive a new problem-dependent regret bound for Thompson Sampling with Gaussian priors (Algorithm 2 in [Agrawal and Goyal, 2017]), a classical stochastic bandit algorithm that has demonstrated excellent empirical performance and has been widely deployed in real-world applications. The existing regret bound is $\sum\limits_{i \in \mathcal{A}: \Delta_i >0}\frac{288 \left(e^{64}+6 \right) \ln \left(T\Delta_i^2 + e^{32} \right)}{\Delta_i} + \frac{10.5}{\Delta_i} + \Delta_i$, where $\mathcal{A}$ is the arm set, $\Delta_i$ denotes the single round performance loss when pulling a sub-optimal arm $i$ instead of the optimal arm, and $T$ is the learning time horizon. Since real-world learning tasks care about the performance with a finite learning time horizon $T$, the existing regret bound is only non-vacuous when $T > 288 \cdot e^{64}$, which may not be practical. Our new regret bound is $\sum\limits_{i \in \mathcal{A}: \Delta_i >0} \frac{1252 \ln \left(T \Delta_i^2 + 100^{\frac{1}{3}}\right)}{\Delta_i} +\frac{18 \ln \left(T\Delta_i^2 \right)}{\Delta_i} + \frac{182.5}{\Delta_i}+ \Delta_i$, which tightens the leading term's coefficient significantly. Despite having made some improvements, we would like to emphasize that the goal of this work is to deepen the understanding of Thompson Sampling from a theoretical perspective to unlock the full potential of this classical learning algorithm for solving challenging real-world learning problems.