SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Similarly, let  $\delta = \varepsilon_1^{1/p}$ . When  $b \in [1 - \delta, 1 + \delta]$ , (39) will trivially hold for any  $\varepsilon_2 > 0$ . Also,  $f(b)$  is bounded on  $\mathbb{R} \backslash [1 - \delta, 1 + \delta]$  and thus there exists finite  $\varepsilon_2$  determined by  $p, \delta$  such that (39) holds.
D.2 Main Lemmas
Lemma D.1. For any  $t \in (kh, (k + 1)h)$ ,  $\partial_t D_{\mathrm{KL}}(\mu_t \| \pi) = -\mathbb{E}_{\mu_t} \left\langle \nabla \log \frac{\pi}{\mu_t}, \mathbb{E}[v_k(X_{kh}) | X_t = \cdot] \right\rangle$
Proof. Let  $\mu_{t|\mathcal{F}_{kh}}$  denote the law of  $X_{t}$  conditioned on the filtration  $\mathcal{F}_{kh}$  at time  $kh$ . Then by Fokker-Planck equation, we have
$$
\partial_ {t} \mu_ {t | \mathcal {F} _ {k h}} = - \operatorname {d i v} \left(\mu_ {t | \mathcal {F} _ {k h}} v _ {k} \left(X _ {k h}\right)\right).
$$
Then we take expectation of the above equation; by Bayesian formula (Vempala & Wibisono, 2019),
$$
\begin{array}{l} \partial_ {t} \mu_ {t} = - \operatorname {d i v} \mathbb {E} \left[ \mu_ {t \mid \mathcal {F} _ {k h}} v _ {k} \left(X _ {k h}\right) \right] \\ = - \operatorname {d i v} \left(\mu_ {t} \mathbb {E} \left[ v _ {k} \left(X _ {k h}\right) \mid X _ {t} = \cdot \right]\right) \\ \end{array}
$$
$$
\partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) = - \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \mathbb {E} \left[ v _ {k} \left(X _ {k h}\right) \mid X _ {t} = \cdot \right] \right\rangle .
$$
Lemma D.2. Suppose that \( h < \frac{1}{G_2} \). Under Assumption 3, for any \( t \in [kh, (k + 1)h] \), \( q >
$$
1, \frac {1}{p} + \frac {1}{q} = 1,
$$
$$
\left. \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla \log \frac {\mu_ {t}}{\mu_ {k h}} \right| \Bigg | _ {q} ^ {q} \leq \left(M _ {p} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {q}
$$
Proof. Note that  $\log \frac{\mu_t}{\mu_{kh}} = -\log \det \left(I_d + (t - kh)\nabla v_k\right)$  since  $x \mapsto x + (t - kh)v_k(x)$  is an orientation-preserving diffeomorphism under  $h < \frac{1}{G_2}$ . Then the following holds:
$$
\begin{array}{l} \left| \left| \nabla \log \frac {\mu_ {t}}{\mu_ {k h}} (x) \right| \right| _ {q} = \sup  _ {\| z \| _ {p} = 1} \left\langle \nabla \log \frac {\mu_ {t}}{\mu_ {k h}} (x), z \right\rangle \\ = \sup  _ {\| z \| _ {p} = 1} \lim  _ {\delta \rightarrow 0} \left| \log \det  \left(I _ {d} + (t - k h) \nabla v _ {k} (x + \delta z)\right) - \log \det  \left(I _ {d} + (t - k h) \nabla v _ {k} (x)\right)\right| \\ \leq \sup  _ {\| z \| _ {p} = 1} \lim  _ {\delta \rightarrow 0} (t - k h) \| \nabla v _ {k} (x + \delta z) - \nabla v _ {k} (x) \| _ {2} (1 - h G _ {2}) ^ {- 1} d \\ \leq M _ {p} (1 - h G _ {2}) ^ {- 1} d h. \\ \end{array}
$$
Here the first equation is due to Young's inequality and the second equation follows the definition of gradient. The inequality in the third line is due to Lemma D.13 and the last one is by Assumption 3. With this uniform bound we finish the proof.
Lemma D.3. Under Assumption 1, 3, and the same conditions in Lemma D.2,
$$
\mathbb {E} _ {\mu_ {t}} \left\| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right\| _ {p} ^ {p} \leq c _ {1} \mathbb {E} _ {\mu_ {t}} \left\| \nabla \log \frac {\pi}{\mu_ {t}} \right\| _ {q} ^ {q} + c _ {2} \left(M _ {p} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {q}
$$
where  $c_{1}, c_{2}$  are defined as:
$$
\left(c _ {1}, c _ {2}\right) = \left\{ \begin{array}{l l} (0, 2 ^ {p - q}) & i f q \leq 2, \\ \left(3 ^ {- p}, \min  \left\{3 ^ {q - p} \left(\frac {p}{q}\right) ^ {\frac {q - p}{q - 1}} \left(1 - \frac {p}{q}\right) ^ {\frac {q - p}{p}}, (q - 1) ^ {p} \left(\frac {\left(\frac {4}{3}\right) ^ {\frac {1}{q - 1}}}{\left(\frac {4}{3}\right) ^ {\frac {1}{q - 1}} - 1}\right) ^ {q - p} \right\}\right) & o t h e r w i s e \end{array} \right. \tag {40}
$$
Proof. Since  $g^{*}(x) = \frac{1}{q}\| x\|_{q}^{q}$ , we have  $\nabla g^{*}(x) = \operatorname{sgn}(x) \odot |x|^{q - 1}$ . Here  $\odot$  means entry-wise product. Apply Lemma D.14 entry-wise and thus
$$
\begin{array}{l} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right| \right| _ {p} ^ {p} \leq c _ {1} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + c _ {2} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\mu_ {t}}{\mu_ {k h}} \right| \right| _ {q} ^ {q} \\ \leq c _ {1} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + c _ {2} \left(M _ {p} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {q}. \\ \end{array}
$$
The second inequality is due to Lemma D.2.
Lemma D.4. Under Assumption 2, 3 and the same conditions in Lemma D.2,
$$
\left. \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}}\right) - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}}\right) \right| _ {2} ^ {2} \leq \beta^ {2} \left(M _ {2} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {2}
$$
Proof. Note that  $g^{*}$  is  $\beta$ -smooth and by Lemma D.2,
$$
\begin{array}{l} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right| \right| _ {2} ^ {2} \leq \beta^ {2} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\mu_ {t}}{\mu_ {k h}} \right| \right| _ {2} ^ {2} \\ \leq \beta^ {2} \left(M _ {2} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {2}. \\ \end{array}
$$
Lemma D.5. Suppose that  $h < \min \left\{\frac{1}{4G_p}, \frac{1}{G_2}\right\}$ . Under Assumption 1, 3, for any  $t \in [kh, (k + 1)h]$ ,
$$
\begin{array}{l} \left. \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \Bigg | _ {p} ^ {p} \\ \leq \frac {2 ^ {p - 1} (1 - h G _ {2}) ^ {- d}}{1 - (4 G _ {p} h) ^ {p}} \varepsilon_ {k} + \frac {2 ^ {p - 1} (4 G _ {p} h) ^ {p}}{1 - (4 G _ {p} h) ^ {p}} \left( \right.\left. (1 + c _ {1}) \mathbb {E} _ {\mu_ {t}} \right|\left| \nabla \log \frac {\pi}{\mu_ {t}} \right|\left. \right| _ {q} ^ {q} + c _ {2} \left(M _ {p} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {q}\left. \right), \\ \end{array}
$$
where  $c_{1}, c_{2}$  are defined in (40).
Proof. By Jensen's inequality,
$$
\begin{array}{l} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \right| _ {p} ^ {p} \\ \leq 2 ^ {p - 1} \left\{\mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - v _ {k} \right| \right| _ {p} ^ {p} + \mathbb {E} _ {\mu_ {t}} \left| \left| v _ {k} - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \right| _ {p} ^ {p} \right\} \tag {41} \\ \leq 2 ^ {p - 1} \left\{\mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - v _ {k} \right| \right| _ {p} ^ {p} + \mathbb {E} \right| \left| v _ {k} (X _ {t}) - v _ {k} (X _ {k h}) \right| \Bigg | _ {p} ^ {p} \Bigg \}. \\ \end{array}
$$
Under Assumption 3 and Jensen's inequality,
$$
\begin{array}{l} \mathbb {E} \left| \left| v _ {k} (X _ {t}) - v _ {k} (X _ {k h}) \right| \right| _ {p} ^ {p} \\ \leq \left(G _ {p}\right) ^ {p} \mathbb {E} \| X _ {t} - X _ {k h} \| _ {p} ^ {p} \\ \leq \left(G _ {p} h\right) ^ {p} \mathbb {E} \left| \left| v _ {k} \left(X _ {k h}\right) \right| \right| _ {p} ^ {p} \\ \leq (G _ {p} h) ^ {p} 4 ^ {p - 1} \left\{\mathbb {E} \left| \left| v _ {k} \left(X _ {k h}\right) - v _ {k} \left(X _ {t}\right) \right| \right| _ {p} ^ {p} + \mathbb {E} \left| \left| v _ {k} \left(X _ {t}\right) - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}} \left(X _ {t}\right)\right) \right| \right| _ {p} ^ {p} \right. \\ + \mathbb {E} \left\| \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}} (X _ {t})\right) - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}} (X _ {t})\right) \right\| _ {p} ^ {p} + \mathbb {E} \left\| \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}} (X _ {t})\right) \right\| _ {p} ^ {p} \}. \\ \end{array}
$$
Rearrange the above inequality and thus
$$
\begin{array}{l} \mathbb {E} \left| \left| v _ {k} (X _ {t}) - v _ {k} (X _ {k h}) \right| \right| _ {p} ^ {p} \\ \leq \left(1 - \left(4 G _ {p} h\right) ^ {p}\right) ^ {- 1} \left(4 G _ {p} h\right) ^ {p} \left\{\mathbb {E} _ {\mu_ {t}} \left| \left| v _ {k} - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}}\right) \right| \right| _ {p} ^ {p} \right. \tag {42} \\ \left. \right. + \mathbb {E} _ {\mu_ {t}} \left| \right.\left| \right. \nabla g ^ {*} \nabla \log \frac {\pi}{\mu_ {k h}}\left. \right) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) \left. \right|\left. \right| _ {p} ^ {p} + \mathbb {E} _ {\mu_ {t}} \left|\left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) \right|\right| _ {p} ^ {p} \left. \right\}. \\ \end{array}
$$
Again note that  $\frac{\mu_t}{\mu_{kh}} = \operatorname*{det}\left(I_d + (t - kh)\nabla v_k\right)^{-1}$ , so
$$
\sup  _ {x} \frac {\mu_ {t}}{\mu_ {k h}} (x) \leq \left(\| (I _ {d} + (t - k h) \nabla v _ {k} (x)) ^ {- 1} \| _ {2}\right) ^ {d} \leq (1 - h G _ {2}) ^ {- d}.
$$
Then by Assumption 1,
$$
\left. \mathbb {E} _ {\mu_ {t}} \right| \left| v _ {k} - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right| \Bigg | _ {p} ^ {p} \leq (1 - h G _ {2}) ^ {- d} \mathbb {E} _ {\mu_ {k h}} \Bigg | \left| v _ {k} - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right| \Bigg | _ {p} ^ {p} \leq (1 - h G _ {2}) ^ {- d} \varepsilon_ {k}. \tag {43}
$$
Combining Lemma D.3 with (42),(43) and plugging them into (41), we finish the proof.
Lemma D.6. Suppose that  $h < \frac{1}{4G_2}$ . Under Assumption 2, 3, for any  $t \in [kh, (k + 1)h]$ ,
$$
\begin{array}{l} \left. \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \Bigg | _ {2} ^ {2} \\ \leq \frac {2 (1 - h G _ {2}) ^ {- d}}{1 - (4 G _ {2} h) ^ {2}} \varepsilon_ {k} + \frac {2 (4 G _ {2} h) ^ {2} \beta^ {2}}{1 - (4 G _ {2} h) ^ {2}} \left(\mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {2} ^ {2} + \left(M _ {2} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {2}\right). \\ \end{array}
$$
Proof. The procedure is exactly the same with Lemma D.5. The only difference appears when applying Lemma D.4 instead of Lemma D.3 in the last step.  $\square$
Lemma D.7. Suppose that  $h \leq \min \left\{\frac{1}{36G_p}, \frac{1 - 2^{-\frac{1}{q}}}{G_2}\right\}$ . Under Assumption 1, 3, for any  $t \in (kh, (k + 1)h)$ ,
$$
\partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) \leq - \frac {1}{1 2} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + A _ {1} \left(M _ {p} d h\right) ^ {q} + A _ {2} \left(1 - h G _ {2}\right) ^ {- d} \varepsilon_ {k},
$$
where  $A_{1}, A_{2}$  are constants only depending on  $q$ :
$$
\left(A _ {1}, A _ {2}\right) = \left\{ \begin{array}{l l} \left(\frac {2 ^ {p + 2}}{p}, 2 ^ {p}\right) & \text {i f} q \leq 2, \\ \left(\frac {7 c _ {2}}{p}, \frac {3}{p}\right) & \text {o t h e r w i s e .} \end{array} \right. \tag {44}
$$
Here  $c_{2}$  is defined in (40).
Proof. By Lemma D.1 and Young's inequality, for any  $\lambda_1, \lambda_2 > 0$ ,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) = - \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}}\right) \right\rangle \\ + \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right\rangle \\ + \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right\rangle \\ \leq - \left(1 - \frac {1}{q} \lambda_ {1} ^ {q} - \frac {1}{q} \lambda_ {2} ^ {q}\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} \\ \left. + \frac {1}{p} \lambda_ {1} ^ {- p} \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}}\right) - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}}\right) \right| \Bigg | _ {p} ^ {p} \\ \left. + \frac {1}{p} \lambda_ {2} ^ {- p} \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \Bigg | _ {p} ^ {p}. \\ \end{array}
$$
Then we apply Lemma D.3 and Lemma D.5,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) \leq - \left(1 - \frac {1}{q} \lambda_ {1} ^ {q} - \frac {1}{p} c _ {1} \lambda_ {1} ^ {- p} - \frac {1}{q} \lambda_ {2} ^ {q} - \frac {1}{p} \lambda_ {2} ^ {- p} \frac {2 ^ {p - 1} (4 G _ {p} h) ^ {p}}{1 - (4 G _ {p} h) ^ {p}} \left(1 + c _ {1}\right)\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} \\ + \frac {c _ {2}}{p} \left(\lambda_ {1} ^ {- p} + \lambda_ {2} ^ {- p} \frac {2 ^ {p - 1} (4 G _ {p} h) ^ {p}}{1 - (4 G _ {p} h) ^ {p}}\right) \left(M _ {p} (1 - h G _ {2}) ^ {- 1} d h\right) ^ {q} + \frac {\lambda_ {2} ^ {- p}}{p} \frac {2 ^ {p - 1} (1 - h G _ {2}) ^ {- d}}{1 - (4 G _ {p} h) ^ {p}} \varepsilon_ {k}. \\ \end{array}
$$
If  $q > 2$  so that  $c_{1} = 3^{-p} < 1$ , take  $\lambda_{1} = c_{1}^{\frac{1}{p + q}}$ ,  $\lambda_{2} = 1$ . Note that for  $h \leq \frac{1}{36G_p}$ ,  $\frac{2^{p - 1}(4G_ph)^p}{1 - (4G_ph)^p} (1 + c_{1}) \leq \frac{4G_{p}h}{1 - 4G_{p}h} \cdot \frac{4}{3} \leq \frac{1}{6}$ . And thus,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) \leq - \left(\frac {2}{3} - \frac {1}{q} - \frac {1}{p} \cdot \frac {1}{6}\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + \frac {7 c _ {2}}{p} (M _ {p} d h) ^ {q} + \frac {3}{p} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k} \\ \leq - \left(\frac {5}{6 p} - \frac {1}{3}\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + \frac {7 c _ {2}}{p} (M _ {p} d h) ^ {q} + \frac {3}{p} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k} \\ \leq - \frac {1}{1 2} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + \frac {7 c _ {2}}{p} (M _ {p} d h) ^ {q} + \frac {3}{p} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k}. \\ \end{array}
$$
If  $q \leq 2$  so that  $c_{1} = 0$ , take  $\lambda_{1} = \lambda_{2} = \left(\frac{q}{3}\right)^{\frac{1}{q}}$ . Note that for  $h \leq \frac{1}{36G_p}$ ,  $\frac{2^{p-1}(4G_ph)^p}{1-(4G_ph)^p}(1+c_1) \leq \frac{(8G_ph)^2}{2(1-4G_ph)} \leq \frac{12}{5}G_ph \leq \frac{1}{15}$ . And thus,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} (\mu_ {t} \| \pi) \leq - \left(\frac {1}{3} - \frac {4}{3 p} (\frac {3}{q}) ^ {\frac {q}{p}} G _ {p} h\right) \mathbb {E} _ {\mu_ {t}} \left\| \nabla \log \frac {\pi}{\mu_ {t}} \right\| _ {q} ^ {q} + \frac {c _ {2} q}{p} (\frac {3}{q}) ^ {q} (M _ {p} d h) ^ {q} + \frac {2 ^ {p}}{p} (\frac {3}{p}) ^ {\frac {q}{p}} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k} \\ \leq - \left(\frac {1}{3} - \frac {4}{p} G _ {p} h\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + \frac {2 ^ {p + 2}}{p} (M _ {p} d h) ^ {q} + 2 ^ {p} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k} \\ \leq - \frac {1}{6} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} + \frac {2 ^ {p + 2}}{p} (M _ {p} d h) ^ {q} + 2 ^ {p} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k}. \\ \end{array}
$$
Therefore, define  $A_{1}, A_{2}$  as in (44) and we finish the proof.
Lemma D.8. Suppose that  $h \leq \frac{1}{4G_2\sqrt{12\kappa^2 + 1}}$ , where  $\kappa \coloneqq \frac{\beta}{\alpha} \geq 1$ . Under Assumption 2, 3, for any  $t \in (kh, (k + 1)h)$ ,
$$
\partial_ {t} D _ {\mathrm {K L}} (\mu_ {t} \| \pi) \leq - \frac {\alpha}{6} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {2} ^ {2} + \frac {3}{\alpha} (\beta M _ {2} d h) ^ {2} + \frac {4}{\alpha} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k}.
$$
Proof. Similar to Lemma D.7, under  $g^{*}$  is  $\alpha$ -strongly convex,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) = - \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}}\right) \right\rangle \\ + \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {t}}) - \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) \right\rangle \\ + \mathbb {E} _ {\mu_ {t}} \left\langle \nabla \log \frac {\pi}{\mu_ {t}}, \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right\rangle \\ \leq - \left(\alpha - \frac {1}{2} \lambda_ {1} ^ {2} - \frac {1}{2} \lambda_ {2} ^ {2}\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {2} ^ {2} \\ \left. + \frac {1}{2} \lambda_ {1} ^ {- 2} \mathbb {E} _ {\mu_ {t}} \right| \left| \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {t}}\right) - \nabla g ^ {*} \left(\nabla \log \frac {\pi}{\mu_ {k h}}\right) \right| \Bigg | _ {2} ^ {2} \\ \left. + \frac {1}{2} \lambda_ {2} ^ {- 2} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla g ^ {*} (\nabla \log \frac {\pi}{\mu_ {k h}}) - \mathbb {E} [ v _ {k} (X _ {k h}) | X _ {t} = \cdot ] \right| \right| _ {2} ^ {2}. \right. \\ \end{array}
$$
Then we apply Lemma D.4 and Lemma D.6,
$$
\begin{array}{l} \partial_ {t} D _ {\mathrm {K L}} \left(\mu_ {t} \| \pi\right) \leq - \left(\alpha - \frac {1}{2} \lambda_ {1} ^ {2} - \frac {1}{2} \lambda_ {2} ^ {2} - \lambda_ {2} ^ {- 2} \frac {(4 G _ {2} h) ^ {2} \beta^ {2}}{1 - (4 G _ {2} h) ^ {2}}\right) \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {2} ^ {2} \\ + \frac {1}{2} \left(\lambda_ {1} ^ {- 2} + \lambda_ {2} ^ {- 2} \frac {2 (4 G _ {2} h) ^ {2}}{1 - (4 G _ {2} h) ^ {2}}\right) (\beta M _ {2} (1 - h G _ {2}) ^ {- 1} d h) ^ {2} + \frac {\lambda_ {2} ^ {- 2}}{2} \frac {2 (1 - h G _ {2}) ^ {- d}}{1 - (4 G _ {2} h) ^ {2}} \varepsilon_ {k}. \\ \end{array}
$$
Take  $\lambda_1 = \lambda_2 = \sqrt{\frac{\alpha}{2}}$ . Note that for  $4G_{2}h \leq \frac{1}{\sqrt{12\kappa^{2} + 1}}$ , we have  $\frac{(4G_2h)^2}{1 - (4G_2h)^2} \leq \frac{1}{12\kappa^2}$ . And thus,
D.3 Proof of Main Results
Theorem D.9. Under Assumption 1, 3, for any step size  $h \leq \min \left\{\frac{1}{36G_p}, \frac{1 - 2^{-\frac{1}{q}}}{G_2}\right\}$ , it holds that
$$
\frac {1}{N h} \int_ {0} ^ {N h} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} d t \leq 1 2 \left(\frac {D _ {\mathrm {K L}} (\mu_ {0} \| \pi)}{N h} + A _ {1} (M _ {p} d h) ^ {q} + A _ {2} (1 - h G _ {2}) ^ {- d} \frac {\sum_ {k = 0} ^ {N - 1} \varepsilon_ {k}}{N}\right),
$$
where  $A_{1}, A_{2}$  defined in (44) are constants that only depend on  $q$ .
Additionally, if  $D_{\mathrm{KL}}(\mu_0\| \pi)\leq K_0$  then for  $N\gtrsim \frac{K_0\left(G_p\vee(qG_2)\right)^{q + 1}}{qA_1(Mpd)^q}$ , we can choose  $h\asymp \left(\frac{K_0}{qA_1(Mpd)^qN}\right)^{\frac{1}{q + 1}}\wedge \frac{1}{dG_2}$ . The following bound holds:
$$
\mathbb {E} _ {\bar {\mu} _ {N h}} \left| \left| \nabla \log \frac {\pi}{\bar {\mu} _ {N h}} \right| \right| _ {q} ^ {q} = \tilde {\mathcal {O}} \left(\left(\frac {M _ {p} K _ {0} d}{N}\right) ^ {\frac {q}{q + 1}} + \frac {G _ {2} K _ {0} d}{N} + \frac {\sum_ {k = 0} ^ {N - 1} \varepsilon_ {k}}{N}\right).
$$
Proof. Under Lemma D.7, take integral of both sides from  $kh$  to  $(k + 1)h$  and we obtain
$$
D _ {\mathrm {K L}} \left(\mu_ {(k + 1) h} \| \pi\right) - D _ {\mathrm {K L}} \left(\mu_ {k h} \| \pi\right) \leq - \frac {1}{1 2} \int_ {k h} ^ {(k + 1) h} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {q} ^ {q} d t + A _ {1} \left(M _ {p} d h\right) ^ {q} h + A _ {2} \left(1 - h G _ {2}\right) ^ {- d} \varepsilon_ {k} h.
$$
Rearranging it and summing from 0 to  $N - 1$
$$
\left. \frac {1}{N h} \int_ {0} ^ {N h} \mathbb {E} _ {\mu_ {t}} \right| \left\| \nabla \log \frac {\pi}{\mu_ {t}} \right\| _ {q} ^ {q} d t \leq 1 2 \left(\frac {D _ {\mathrm {K L}} (\mu_ {0} \| \pi)}{N h} + A _ {1} (M _ {p} d h) ^ {q} + A _ {2} (1 - h G _ {2}) ^ {- d} \frac {\sum_ {k = 0} ^ {N - 1} \varepsilon_ {k}}{N}\right). \tag {45}
$$
Note that for any convex function  $g^{*}$  on  $\mathbb{R}^d$ ,  $(a,b)\mapsto g^{*}(a / b)b$  is also convex on  $\mathbb{R}^d\times \mathbb{R}_+$ . Therefore,  $\mu \mapsto \mathbb{E}_{\mu}g^{*}\bigl (\nabla \log \frac{\pi}{\mu}\bigr)$  is convex in the classical sense on the space of probability measures. And thus
$$
\left. \mathbb {E} _ {\bar {\mu} _ {N h}} \right| \left| \nabla \log \frac {\pi}{\bar {\mu} _ {N h}} \right| \Bigg | _ {q} ^ {q} \leq \frac {1}{N h} \int_ {0} ^ {N h} \mathbb {E} _ {\mu_ {t}} \left| \right| \nabla \log \frac {\pi}{\mu_ {t}} \Bigg | \Bigg | _ {q} ^ {q} d t. \tag {46}
$$
We finish the proof by plugging the step size  $h$  in (45) and hiding all the constants that only depend on  $q$ .
Theorem D.10. Under Assumption 2, 3, for any step size  $h \leq \frac{1}{4G_2\sqrt{12\kappa^2 + 1}}$ , where  $\kappa := \frac{\beta}{\alpha} \geq 1$  it holds that
$$
\frac {1}{N h} \int_ {0} ^ {N h} \mathbb {E} _ {\mu_ {t}} \left| \left| \nabla \log \frac {\pi}{\mu_ {t}} \right| \right| _ {2} ^ {2} d t \leq \frac {6}{\alpha} \left(\frac {D _ {\mathrm {K L}} (\mu_ {0} \| \pi)}{N h} + \frac {3}{\alpha} \beta^ {2} M _ {2} ^ {2} d ^ {2} h ^ {2} + \frac {4}{\alpha} (1 - h G _ {2}) ^ {- d} \frac {\sum_ {k = 0} ^ {N - 1} \varepsilon_ {k}}{N}\right).
$$
For simplicity, assume  $\alpha = 1$ . If and  $D_{\mathrm{KL}}(\mu_0\| \pi)\leq K_0$ , then we can choose  $h\asymp (\frac{K_0}{(\kappa M_2d)^2N})^{\frac{1}{3}}\wedge \frac{1}{dG_2}\wedge \frac{1}{\kappa G_2}$ . The following bound holds:
$$
\mathbb {E} _ {\bar {\mu} _ {N h}} \bigg \| \nabla \log \frac {\pi}{\bar {\mu} _ {N h}} \bigg \| _ {2} ^ {2} = \mathcal {O} \left((\frac {\kappa M _ {2} K _ {0} d}{N}) ^ {\frac {2}{3}} + \frac {G _ {2} K _ {0} (d + \kappa)}{N} + \frac {\sum_ {k = 0} ^ {N - 1} \varepsilon_ {k}}{N}\right).
$$
Proof. Under Lemma D.8, take integral of both sides from  $kh$  to  $(k + 1)h$  and we obtain
$$
D _ {\mathrm {K L}} (\mu_ {(k + 1) h} \| \pi) - D _ {\mathrm {K L}} (\mu_ {k h} \| \pi) \leq - \frac {\alpha}{6} \int_ {k h} ^ {(k + 1) h} \mathbb {E} _ {\mu_ {t}} \left\| \nabla \log \frac {\pi}{\mu_ {t}} \right\| _ {2} ^ {2} d t + \frac {3}{\alpha} \beta^ {2} M _ {2} ^ {2} d ^ {2} h ^ {3} + \frac {4}{\alpha} (1 - h G _ {2}) ^ {- d} \varepsilon_ {k} h.
$$
The remaining part is similar to the proof of Theorem D.9.
D.4 Discussions
The convergence of score divergence only guarantees that the particle distribution gets the local structure of  $\pi$  correct (Balasubramanian et al., 2022). To obtain a stronger convergence guarantee, we still need isoperimetry condition of target distribution. We start with  $L_{2}$ -GF.
Theorem D.11. If we additionally assume that  $\pi$  satisfies log-Sobolev inequality with constant  $\lambda$ , i.e.
$$
E n t _ {\pi} (f ^ {2}) \leq \frac {2}{\lambda} \mathbb {E} _ {\pi} [ \| \nabla f \| _ {2} ^ {2} ], \text {f o r a l l s m o o t h} f: \mathbb {R} ^ {d} \rightarrow \mathbb {R}.
$$
then under the same conditions of Theorem D.10 with  $\alpha = \beta = 1, \varepsilon_k \leq \epsilon$ , it holds that
$$
D _ {\mathrm {K L}} \left(\mu_ {N h} \| \pi\right) \leq \exp \left(- \frac {\lambda N h}{3}\right) D _ {\mathrm {K L}} \left(\mu_ {0} \| \pi\right) + 3 \left(3 \left(M _ {2} d h\right) ^ {2} + 4 \left(1 - h G _ {2}\right) ^ {- d} \epsilon\right) \lambda^ {- 1}.
$$
In particular, if  $\epsilon \lesssim (\frac{M_2}{G_2})^2$ , we take  $h \asymp \frac{\sqrt{\epsilon}}{M_2d}$  and then we obtain the guarantee  $D_{\mathrm{KL}}(\mu_{Nh} \| \pi) \lesssim \lambda^{-1} \epsilon$  after