$$
\begin{array}{l} L _ {\text {n a i v e}} \left(\boldsymbol {X} ^ {(i)}\right) \\ = -\gamma g(\boldsymbol{x}^{(i)},\boldsymbol{x}_{*}^{(i)}) + \frac{1}{N - 1}\sum_{\substack{j = 1\\ j\neq i}}^{N}\gamma g(\boldsymbol{x}^{(i)},\boldsymbol{x}_{*}^{(j)}) \\ = - \frac {N}{N - 1} \left(\frac {N - 1}{N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) - \frac {1}{N} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) (26) \\ = - \frac {N}{N - 1} \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) - \frac {1}{N} \sum_ {j = 1} ^ {N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) (27) \\ = - \frac {N}{N - 1} \left[ \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) - \log \exp \left(\frac {1}{N} \sum_ {j = 1} ^ {N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) \right] (28) \\ \leq - \frac {N}{N - 1} \left[ \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) - \log \left(\frac {1}{N} \sum_ {j = 1} ^ {N} \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right)\right) \right] (29) \\ = - \frac {N}{N - 1} \left(\log \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}}{\sum_ {j = 1} ^ {N} \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)}} + \log N\right) (30) \\ = - \frac {N}{N - 1} \log \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}}{\sum_ {j = 1} ^ {N} \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)}} - \frac {N \log N}{N - 1}, (31) \\ L _ {\text {n a i v e}} \left(\boldsymbol {X} ^ {(i)}\right) \\ = -\gamma g(\boldsymbol{x}^{(i)},\boldsymbol{x}_{*}^{(i)}) + \frac{1}{N - 1}\sum_{\substack{j = 1\\ j\neq i}}^{N}\gamma g(\boldsymbol{x}^{(i)},\boldsymbol{x}_{*}^{(j)}) \\ = \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) + \frac {1}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) - 2 \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) (32) \\ = 2 \left[ \frac {1}{2} \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) + \frac {1}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) - \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) \right] (33) \\ = 2 \left\{\log \exp \left[ \frac {1}{2} \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right) + \frac {1}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)\right) \right] - \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) \right\} (34) \\ \end{array}
$$