we can always work on a scale-transformation of $\xi$, $\xi'=c\xi$, resulting in a transformed $V$, $V'(x)=V(x/c)+\log c$. In what follows we adopt this simplification, so that we have $V(x)\ge0$.

\[assu:v\_dissipative\]There exist constants $\alpha>0$ and $0<\beta<1$ such that for all $x\in\mathbb{R}^{r}$, $$\frac{1}{2}\langle\nabla V(x),x\rangle\ge\beta V(x)+\gamma^{2}C_{\beta}\Vert x\Vert^{2}-\alpha,\quad C_{\beta}=\frac{\beta(2-\beta)}{8(1-\beta)}.$$

Assumption \[assu:v\_dissipative\] is a common and standard regularity condition on $V$ coming from @mattingly2002ergodicity. We then have the following conclusion:

\[thm:finite\_moments\]Suppose Assumptions \[assu:v\_boundedness\] and \[assu:v\_dissipative\] hold, and choose $\delta$ small enough such that $\delta\le\min\{1/\gamma,\gamma/(2\nu),(D+1-\sqrt{D^{2}+1})/\gamma\}$, $D=\gamma^{4}C_{\beta}/\nu^{2}$. Then for any fixed $l>0$ and all $k\ge0$, there exist constants $C=C(l,\delta)>0$, $\lambda=\lambda(l,\delta)>0$, and an integer $m=m(l)>0$ such that $$\mathbb{E}\left(\Vert\xi_{k}\Vert^{2l}+\Vert\rho_{k}\Vert^{2l}\right)\le C\left\{ 1+\left(\Vert\xi_{0}\Vert^{m}+\Vert\rho_{0}\Vert^{m}\right)e^{-\lambda k}\right\} .$$

Next, we present the main result for the underdamped Langevin algorithm. Let $\varphi:\mathbb{R}^{2r}\to\mathbb{R}$ be a multivariate function with the notation $\varphi(w)\equiv\varphi(\xi,\rho)$, where $w=(\xi^{\mathrm{T}},\rho^{\mathrm{T}})^{\mathrm{T}}$. Then define its expectation with respect to $\pi_{W}$ as $\bar{\varphi}=\mathbb{E}_{\pi_{W}}\varphi\coloneqq\int\varphi(\xi,\rho)\pi_{W}(\xi,\rho)\mathrm{d}\xi\mathrm{d}\rho$. It is easy to see that if $\varphi(w)=\nabla f(\theta;\xi)$, then $\bar{\varphi}=\mathbb{E}_{\pi_{W}}\varphi=\nabla F(\theta)$ is exactly the true gradient function we are interested in. Driven by the motivation to approximate $\bar{\varphi}$, Theorem \[thm:control\_bias\_mse\] below shows that we can construct an estimator $\hat{\varphi}$ using the sequence $\{W_{k}\}$, where $\hat{\varphi}=K^{-1}\sum_{k=0}^{K-1}\varphi(W_{k})$.

\[thm:control\_bias\_mse\]Let $\varphi$, $\bar{\varphi}$ , and $\hat{\varphi}$ be defined as above, with $\varphi\in\mathscr{C}_{poly}^{r+5}$. Assume that the conditions in 