Theorem \[thm:finite\_moments\] hold. Then there exist constants $C_{1}>0$ and $C_{2}>0$ such that for any $\delta>0$ in the range and any integer $K>0$, we have $$\begin{aligned}
\left|\mathbb{E}(\hat{\varphi})-\bar{\varphi}\right| & \le C_{1}\left(\frac{1}{K\delta}+\delta\right),\\
\mathbb{E}\left[\left(\hat{\varphi}-\bar{\varphi}\right)^{2}\right] & \le C_{2}\left(\frac{1}{K\delta}+\delta^{2}\right).\end{aligned}$$

Theorem \[thm:control\_bias\_mse\] shows that in general $\hat{\varphi}$ is a biased estimator for $\bar{\varphi}$, but its bias and mean squared error can be made arbitrarily small by appropriately choosing the algorithm parameters $\delta$ and $K$.

Here we make a few remarks about the results in this section. Theorem \[thm:finite\_moments\] is similar to Proposition 2.7 of @kopec2015weak, but they use the implicit Euler scheme to discretize the Langevin SDE, which is computationally much harder. Therefore, Theorem \[thm:finite\_moments\] is a new result for the explicit Euler scheme given by and . The rates in Theorem \[thm:control\_bias\_mse\] are known results [@chen2015convergence]. However, in most prior art the assumptions to make Theorem \[thm:control\_bias\_mse\] hold are highly non-trivial and very difficult to check for real machine learning models. For example, @chen2015convergence needs to assume that our conclusion in Theorem \[thm:finite\_moments\] holds, along with other technical conditions. In contrast, our assumptions are only made on the log-density function $V(\xi)$, which is the actual model that machine learning practitioners are given. In this sense, the 