$$
\begin{array}{l} \leq 2 \left\{\log \left[ \frac {1}{2} \left(\exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) + \exp \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \frac {\left(\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)\right)}{N - 1}\right) \right] - \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) \right\} (35) \\ = 2 \left[ \log \left(\exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) + \exp \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \frac {\left(\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)\right)}{N - 1}\right) - \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) - \log 2 \right] (36) \\ = 2 \log \left(1 + \frac {\exp \left(\sum_ {j = 1} ^ {N} \frac {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)}{N - 1}\right)}{\exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right)}\right) - 2 \log 2, (37) \\ 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 {1}{N - 1} \left(\sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right) + \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) - 2 (N - 1) \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) (38) \\ = \frac {2}{N - 1} \left(\sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \frac {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right) + \gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}{2} - (N - 1) \gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right) (39) \\ = \frac {2}{N - 1} \left(\sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \log \exp \frac {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right) + \gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}{2} - (N - 1) \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right)\right) (40) \\ \leq \frac {2}{N - 1} \left(\sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \log \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)} + \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}}{2} - (N - 1) \log \exp \left(\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)\right)\right) (41) \\ = \frac {2}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \left(\log \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)} + \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}}{2} - \log \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)}\right) (42) \\ = \frac {2}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \log \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(j)}\right)} + \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}}{2 \mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)} , \boldsymbol {x} _ {*} ^ {(i)}\right)}} (43) \\ = \frac {2}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \log \left[ \frac {1}{2} \left(1 + \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)}}{\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)}}\right) \right] (44) \\ = \frac {2}{N - 1} \sum_ {\substack {j = 1 \\ j \neq i}} ^ {N} \log \left(1 + \frac {\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(j)}\right)}}{\mathrm {e} ^ {\gamma g \left(\boldsymbol {x} ^ {(i)}, \boldsymbol {x} _ {*} ^ {(i)}\right)}}\right) - 2 \log 2, (45) \\ \end{array}
$$