SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Here, let  $G(x) = H(x) - H(0)$ . We can see that  $\mu(x)$  can be expressed as  $\mu(x) = \frac{\exp(-G(x))\nu(x)}{Z_G}$  with a normalizing constant  $Z_G > 0$ . Note that, by the Lipschitz continuity assumption of  $H$ , we have
$$
| G (x) | \leq \bar {L} \| x \|.
$$
Hence, we have the following evaluation of the normalizing constant  $Z_{G}$ :
$$
\begin{array}{l} Z _ {G} = \int \exp (- G (x)) \nu (x) d x \\ \geq \int \exp \left(- \bar {L} \| x \| - c _ {2} \left(c _ {V} + \| x \| ^ {2}\right)\right) \frac {1}{\sqrt {\left(2 \pi \left(1 / c _ {1}\right)\right) ^ {d}}} d x \\ \geq \int \exp \left(- \frac {3 c _ {2}}{2} \| x \| ^ {2} - \frac {\bar {L} ^ {2}}{2 c _ {2}} - c _ {2} c _ {V}\right) \frac {1}{\sqrt {(2 \pi (1 / c _ {1})) ^ {d}}} \mathrm {d} x \\ = \left(\frac {c _ {1}}{3 c _ {2}}\right) ^ {d / 2} \exp \left(- \frac {\bar {L} ^ {2}}{2 c _ {2}} - c _ {2} c _ {V}\right). \\ \end{array}
$$
Theorem 2.7 of Cattiaux and Guillin (2022) claims that  $\mu$  satisfies the LSI with a constant  $\alpha$  such that
$$
\frac {2}{\alpha} \leq \frac {(\beta + 1) (1 + \theta^ {- 1})}{\beta} \frac {2}{\alpha^ {\prime}} + \frac {1}{\tilde {\alpha}} (2 + \mathbb {E} _ {\nu} [ G - \log (Z _ {G}) ]) + \bar {L} ^ {2} \frac {2}{\tilde {\alpha} \alpha^ {\prime}} \left(\frac {(1 + \theta) (1 + \beta)}{4 \beta} + \frac {\beta^ {2}}{2}\right),
$$
for any  $\beta > 0$  and  $\theta > 0$ . The right hand side can be bounded by
$$
\begin{array}{l} \frac {(\beta + 1) (1 + \theta^ {- 1})}{\beta} \frac {2}{c _ {1}} + \frac {1}{\tilde {\alpha}} \left(2 + \bar {L} \mathbb {E} _ {\nu} [ \| X \| ] + \frac {\bar {L} ^ {2}}{2 c _ {2}} + \frac {d}{2} \log \left(3 c _ {2} / c _ {1}\right)\right) + \bar {L} ^ {2} \frac {2}{\tilde {\alpha} c _ {1}} \left(\frac {(1 + \theta) (1 + \beta)}{4 \beta} + \frac {\beta^ {2}}{2}\right) \\ \leq \frac {(\beta + 1) (1 + \theta^ {- 1})}{\beta} \frac {2}{c _ {1}} + \frac {1}{\tilde {\alpha}} \left(2 + \bar {L} \sqrt {\frac {d}{c _ {1}}} + \frac {\bar {L} ^ {2}}{2 c _ {1}} + \frac {d}{2} \log \left(3 c _ {2} / c _ {1}\right)\right) + \bar {L} ^ {2} \frac {2}{\tilde {\alpha} c _ {1}} \left(\frac {(1 + \theta) (1 + \beta)}{4 \beta} + \frac {\beta^ {2}}{2}\right) \\ \leq \frac {(\beta + 1) (1 + \theta^ {- 1})}{\beta} \frac {2}{c _ {1}} + \frac {1}{\tilde {\alpha}} \left(2 + \frac {\bar {L} ^ {2}}{c _ {1}} + \frac {d}{2} (\log (3 c _ {2} / c _ {1}) + 1)\right) + \bar {L} ^ {2} \frac {2}{\tilde {\alpha} c _ {1}} \left(\frac {(1 + \theta) (1 + \beta)}{4 \beta} + \frac {\beta^ {2}}{2}\right) \\ \leq \frac {(\beta + 1) (1 + \theta^ {- 1})}{\beta} \frac {2}{c _ {1}} + \frac {1}{\tilde {\alpha}} \left[ 2 + \frac {d}{2} (\log (c _ {2} / c _ {1}) + 2) + \frac {\bar {L} ^ {2}}{c _ {1}} \left(1 + \frac {(1 + \theta) (1 + \beta)}{2 \beta} + \beta^ {2}\right) \right]. \\ \end{array}
$$
Hence, if we set  $\beta = 1$  and  $\theta = 1$ , we have
$$
\begin{array}{l} \frac {1}{\alpha} \leq 2 \frac {2}{c _ {1}} + \frac {1}{2 \tilde {\alpha}} \left[ 2 + \frac {d}{2} \left(\log \left(c _ {2} / c _ {1}\right) + 2\right) + \frac {\bar {L} ^ {2}}{c _ {1}} (1 + 2 + 1) \right], \\ \leq \frac {4}{c _ {1}} + \left(\frac {\bar {L}}{c _ {1}} + \sqrt {\frac {2}{c _ {1}}}\right) ^ {2} e ^ {\frac {\bar {L} ^ {2}}{2 c _ {1}}} \left[ 2 + d + \frac {d}{2} \log \left(\frac {c _ {2}}{c _ {1}}\right) + 4 \frac {\bar {L} ^ {2}}{c _ {1}} \right]. \\ \end{array}
$$
E.4 Uniform Log-Sobolev Inequality
Lemma 7 (Uniform log-Sobolev inequality). Under the same condition as Theorem 2, it holds that
$$
\left. \right. - \frac {\lambda^ {2}}{2 N} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[\left\| \nabla_ {i} \log \left(\frac {\tilde {\mu} _ {t} ^ {(N)}}{p ^ {(N)}} (\mathcal {X})\right)\right\| ^ {2} \right] \leq - \frac {\lambda \alpha}{2} \left(\frac {1}{N} \mathscr {F} ^ {N} \left(\tilde {\mu} _ {t} ^ {(N)}\right) - \mathscr {F} \left(\mu^ {*}\right)\right) + \frac {C _ {\lambda}}{N}, \tag {27}
$$
with a constant  $C_{\lambda} = 2\lambda L\alpha (1 + 2c_{L}\bar{R}^{2}) + 2\lambda^{2}L^{2}\bar{R}^{2}$
Proof. The derivation is analogous to Chen et al. (2022), but we present the full proof for the sake of the completeness. For  $\mathcal{X} = (X_i)_{i=1}^N$ , let  $\mathcal{X}^{-i} \coloneqq (X_j)_{j \neq i}$ . Then, it holds that
$$
\begin{array}{l} - \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\frac {\tilde {\mu} _ {t} ^ {(N)}}{p ^ {(N)}} (\mathcal {X})\right) \right\| ^ {2} \right] \\ = - \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log (\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X}})}{\delta \mu} (X _ {i}) \right\| ^ {2} \right] \\ = - \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})\right) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) + \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \right. \right. \\ \left. - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X}})}{\delta \mu} (X _ {i}) \right\rVert^ {2} \Biggr ] \\ \leq - \frac {1}{2} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})\right) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \right\| ^ {2} \right] \\ + \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X}})}{\delta \mu} (X _ {i}) \right\| ^ {2} \right] \\ \leq - \frac {1}{2} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})\right) - \frac {1}{\lambda} \nabla \frac {\delta F \left(\mu_ {\mathcal {X} ^ {- i}}\right)}{\delta \mu} (X _ {i}) \right\| ^ {2} \right] \\ + \sum_ {i = 1} ^ {N} L ^ {2} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ W _ {2} ^ {2} \left(\mu_ {\mathcal {X}}, \mu_ {\mathcal {X} ^ {- i}}\right) \right] \\ = - \frac {1}{2} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})\right) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \right\| ^ {2} | \mathcal {X} ^ {- i} \right] \right] \\ + \sum_ {i = 1} ^ {N} L ^ {2} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ W _ {2} ^ {2} \left(\mu_ {\mathcal {X}}, \mu_ {\mathcal {X} ^ {- i}}\right) \right], \\ \end{array}
$$
where the second inequality is due to the Lipschitz continuity of  $\nabla \frac{\delta U}{\delta \mu}$  in terms of the Wasserstein distance (Assumption 2). Here, the term corresponding to the Wasserstein distance can be upper bounded as
$$
\begin{array}{l} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ W _ {2} ^ {2} \left(\mu_ {\mathcal {X}}, \mu_ {\mathcal {X} ^ {- i}}\right) \right] \leq \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \frac {1}{N (N - 1)} \sum_ {j \neq i} \| X _ {j} - X _ {i} \| ^ {2} \right] \\ \leq \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \frac {2}{N (N - 1)} \sum_ {j \neq i} \| X _ {j} \| ^ {2} + \frac {2}{N} \| X _ {i} \| ^ {2} \right] \\ \leq \frac {4}{N} \bar {R} ^ {2}. \\ \end{array}
$$
Denote by  $P_{X_i|\mathcal{X}^{-i}}$  the conditional law of  $X_{i}$  conditioned by  $\mathcal{X}^{-i}$  and denote by  $P_{\mathcal{X}^{-i}}$  the marginal law of  $\mathcal{X}^{-i}$  where the joint law of  $\mathcal{X}$  is  $\tilde{\mu}_t^{(N)}$ . Then, it holds that
$$
\begin{array}{l} \nabla_ {i} \log (\tilde {\mu} _ {t} ^ {(N)} (\mathcal {X})) = \frac {\nabla_ {i} (P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i}) P _ {\mathcal {X} ^ {- i}} (\mathcal {X} ^ {- i}))}{P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i}) P _ {\mathcal {X} ^ {- i}} (\mathcal {X} ^ {- i})} = \frac {\nabla_ {i} P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i})}{P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i})} = \nabla_ {i} \log (P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i})). \\ - \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} ^ {- i} \sim P _ {\mathcal {X} ^ {- i}}} \left[ \mathbb {E} _ {X _ {i} \sim P _ {X _ {i} | \mathcal {X} ^ {- i}}} \left[ \left\| \nabla_ {i} \log \left(P _ {X _ {i} | \mathcal {X} ^ {- i}} (X _ {i})\right) - \frac {1}{\lambda} \nabla \frac {\delta F (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \right\| ^ {2} \mid \mathcal {X} ^ {- i} \right] \right] \\ \leq - 2 \alpha \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ D \left(p _ {\mathcal {X} ^ {- i}}, P _ {X _ {i} | \mathcal {X} ^ {- i}}\right) \right], \\ \end{array}
$$
by the LSI condition of the proximal Gibbs measure (Assumption 3). Let  $\nu_{r}$  be the distribution with the density  $\nu_{r} \propto \exp(-r(x)/\lambda)$ . Note that the proximal Gibbs measure is the minimizer of the linearized objective:
$$
\begin{array}{l} p _ {\mathcal {X} ^ {- i}} = \operatorname * {a r g m i n} _ {\mu \in \mathcal {P}} \left\{\int \frac {\delta U (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \mathrm {d} \mu + \mathbb {E} _ {\mu} [ r ] + \lambda \mathrm {E n t} (\mu) \right\} \\ = \operatorname * {a r g m i n} _ {\mu \in \mathcal {P}} \left\{\int \frac {\delta U (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (X _ {i}) \mathrm {d} \mu + \lambda D (\nu_ {r}, \mu) \right\}. \\ \end{array}
$$
Then, by the optimality of the proximal Gibbs measure, it holds that
$$
\begin{array}{l} \lambda D \left(p _ {\mathcal {X} ^ {- i}}, P _ {X _ {i} | \mathcal {X} ^ {- i}}\right) \\ = \int \frac {\delta U (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (x _ {i}) \left(P _ {X _ {i} | \mathcal {X} ^ {- i}} - p _ {\mathcal {X} ^ {- i}}\right) (\mathrm {d} x _ {i}) + \lambda D \left(\nu_ {r}, P _ {X _ {i} | \mathcal {X} ^ {- i}}\right) - \lambda D \left(\nu_ {r}, p _ {\mathcal {X} ^ {- i}}\right) \\ \geq \int \frac {\delta U \left(\mu_ {\mathcal {X} ^ {- i}}\right)}{\delta \mu} \left(x _ {i}\right) \left(P _ {X _ {i} \mid \mathcal {X} ^ {- i}} - \mu^ {*}\right) (\mathrm {d} x _ {i}) + \lambda D \left(\nu_ {r}, P _ {X _ {i} \mid \mathcal {X} ^ {- i}}\right) - \lambda D \left(\nu_ {r}, \mu^ {*}\right). \tag {28} \\ \end{array}
$$
The expectation of the first term of the right hand side can be further evaluated as
$$
\begin{array}{l} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \int \frac {\delta U (\boldsymbol {\mu} _ {\mathcal {X} ^ {- i}})}{\delta \boldsymbol {\mu}} (x _ {i}) (P _ {X _ {i} | \mathcal {X} ^ {- i}} - \boldsymbol {\mu} ^ {*}) (\mathrm {d} x _ {i}) \right] \\ = \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \int \frac {\delta U (\mu_ {\mathcal {X} - i})}{\delta \mu} (x _ {i}) \delta_ {X _ {i}} (\mathrm {d} x _ {i}) - \int \frac {\delta U (\mu_ {\mathcal {X} - i})}{\delta \mu} (x _ {i}) \mu^ {*} (\mathrm {d} x _ {i}) \right] \\ \geq \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left\{\int \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} \left(x _ {i}\right) \delta_ {X _ {i}} \left(\mathrm {d} x _ {i}\right) - \int \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} \left(x _ {i}\right) \mu^ {*} \left(\mathrm {d} x _ {i}\right) \right. \\ \left. - \frac {2 L}{N} \left[ 2 + c _ {L} \left(\| X _ {i} \| ^ {2} + \mathbb {E} _ {X \sim \mu^ {*}} [ \| X \| ^ {2} ] + \| X _ {i} \| ^ {2} + \frac {1}{N - 1} \sum_ {j \neq i} \| X _ {j} \| ^ {2}\right) \right] \right\} \\ \geq N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \int \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} (x) \mu_ {\mathcal {X}} (\mathrm {d} x) - \int \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} (x) \mu^ {*} (\mathrm {d} x) \right] - 2 L \left(2 + 4 c _ {L} \bar {R} ^ {2}\right) \\ \geq N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ U (\mu_ {\mathcal {X}}) - U (\mu^ {*}) - 2 L (2 + 4 c _ {L} \bar {R} ^ {2}) \right] \\ \geq N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ U \left(\mu_ {\mathcal {X}}\right) - U \left(\mu^ {*}\right) \right] - 2 L \left(2 + 4 c _ {L} \bar {R} ^ {2}\right) \\ \geq N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ U \left(\mu_ {\mathcal {X}}\right) - U \left(\mu^ {*}\right) \right] - 4 L \left(1 + 2 c _ {L} \bar {R} ^ {2}\right), \\ \end{array}
$$
where the first inequality is due to Lemma 8 and the second inequality is due to Lemma 1 (we also notice that  $\mathbb{E}_{X\sim \mu^{*}}[\| X\|^{2}]\leq \bar{R}^{2}$ ). The second term in the right hand side of Eq. (28) can be evaluated as
$$
\begin{array}{l} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\tilde {\mu} _ {t} ^ {(N)}} [ D (\nu_ {r}, P _ {X _ {i} | \mathcal {X} ^ {- i}}) ] = \sum_ {i = 1} ^ {N} \mathbb {E} _ {\tilde {\mu} _ {t} ^ {(N)}} [ \operatorname {E n t} (P _ {X _ {i} | \mathcal {X} ^ {- i}}) ] - \mathbb {E} _ {\tilde {\mu} _ {t} ^ {(N)}} \left[ \sum_ {i = 1} ^ {N} \log (\nu_ {r} (X _ {i})) \right] \\ \geq \mathrm {E n t} (\tilde {\mu} _ {t} ^ {(N)}) - \mathbb {E} _ {\tilde {\mu} _ {t} ^ {(N)}} \left[ \sum_ {i = 1} ^ {N} \log (\nu_ {r} (X _ {i})) \right] = D (\nu_ {r} ^ {N}, \tilde {\mu} _ {t} ^ {(N)}), \\ \end{array}
$$
where we used Lemma 3.6 of Chen et al. (2022) in the first inequality. Combining all of them, we arrive at
$$
\begin{array}{l} - \sum_ {i = 1} ^ {N} \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} \left[ \left\| \nabla_ {i} \log \left(\frac {\tilde {\mu} _ {t} ^ {(N)}}{p ^ {(N)}} (\mathcal {X})\right) \right\| ^ {2} \right] \\ \leq - \frac {\alpha}{\lambda} \left[ N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} [ U (\mu_ {\mathcal {X}}) ] + \lambda D (\nu_ {r} ^ {N}, \tilde {\mu} _ {t} ^ {(N)}) - N (U (\mu^ {*}) + \lambda D (\nu_ {r}, \mu^ {*})) - 4 L \left(1 + 2 c _ {L} \bar {R} ^ {2}\right) \right] + 4 L ^ {2} \bar {R} ^ {2} \\ \end{array}
$$
$$
\begin{array}{l} = - \frac {\alpha}{\lambda} \left[ N \mathbb {E} _ {\mathcal {X} \sim \tilde {\mu} _ {t} ^ {(N)}} [ F (\mu_ {\mathcal {X}}) ] + \lambda \mathrm {E n t} (\tilde {\mu} _ {t} ^ {(N)}) - N (F (\mu^ {*}) + \lambda \mathrm {E n t} (\mu^ {*})) - 4 L \left(1 + 2 c _ {L} \bar {R} ^ {2}\right) \right] + 4 L ^ {2} \bar {R} ^ {2} \\ = - \frac {\alpha}{\lambda} \left(\mathcal {F} ^ {N} \left(\tilde {\mu} _ {t} ^ {(N)}\right) - N \mathcal {F} \left(\mu^ {*}\right)\right) + \frac {4 L \alpha}{\lambda} \left(1 + 2 c _ {L} \bar {R} ^ {2}\right) + 4 L ^ {2} \bar {R} ^ {2}. \\ \end{array}
$$
Then, we have the assertion with
$$
C _ {\lambda} = 2 \lambda L \alpha \left(1 + 2 c _ {L} \bar {R} ^ {2}\right) + 2 \lambda^ {2} L ^ {2} \bar {R} ^ {2}.
$$
Lemma 8. With the same setting as Lemma 7, it holds that
$$
\left| \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} (x) - \frac {\delta U (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (x) \right| \leq \frac {L}{N} \left[ 2 + c _ {L} \left(2 \| x \| ^ {2} + \| X _ {i} \| ^ {2} + \frac {1}{N - 1} \sum_ {j \neq i} \| X _ {j} \| ^ {2}\right) \right]
$$
Proof. Let  $\mu_{\mathcal{X},\theta} \coloneqq \theta \mu_{\mathcal{X}} + (1 - \theta) \mu_{\mathcal{X}^{-i}}$ . Then, we see that
$$
\begin{array}{l} \left| \frac {\delta U (\mu_ {\mathcal {X}})}{\delta \mu} (x) - \frac {\delta U (\mu_ {\mathcal {X} ^ {- i}})}{\delta \mu} (x) \right| = \int_ {0} ^ {1} \left| - \frac {1}{N} \frac {\delta^ {2} U (\mu_ {\mathcal {X} , \theta})}{\delta \mu^ {2}} (x, X _ {i}) + \frac {1}{N (N - 1)} \sum_ {j \neq i} \frac {\delta^ {2} U (\mu_ {\mathcal {X} , \theta})}{\delta \mu^ {2}} (x, X _ {j}) \right| d \theta \\ \leq \frac {L}{N} \left[ 2 + c _ {L} \left(2 \| x \| ^ {2} + \| X _ {i} \| ^ {2} + \frac {1}{N - 1} \sum_ {j \neq i} \| X _ {j} \| ^ {2}\right) \right], \\ \end{array}
$$
where we used Assumption 2 (this is the only place where the boundedness of the second order variation of  $U$  is used). Hence, we obtain the assertion.