$$
\begin{array}{l} \sum_ {j = 2} ^ {n} \frac {1}{n} K _ {\mathcal {S} ^ {\prime}} \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j} ^ {\prime}\right) - \sum_ {j = 2} ^ {n} \frac {1}{n} K _ {\mathcal {S} _ {1} ^ {\prime}} \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j} ^ {\prime}\right) \\ = \frac {1}{n} \sum_ {j = 2} ^ {n} \mathrm {K} _ {\mathcal {S} ^ {\prime}} \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j} ^ {\prime}\right) - \mathrm {K} _ {S _ {1} ^ {\prime}} \left(\boldsymbol {z} _ {i}, \boldsymbol {z} _ {j} ^ {\prime}\right) \\ = \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle d t - \int_ {0} ^ {T} \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle d t \\ = \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle - \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle d t \\ = \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle - \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle \\ + \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle - \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle d t \\ = \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(\mathcal {S} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) - \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle \\ + \left\langle \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S ^ {\prime}\right), \boldsymbol {z} _ {i}\right) - \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {i}\right), \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\rangle d t \\ \leq \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} \| \nabla_ {\boldsymbol {w}} \ell (A _ {t} (\mathcal {S} ^ {\prime}), \boldsymbol {z} _ {i}) \| \left\| \nabla_ {\boldsymbol {w}} \ell (A _ {t} (\mathcal {S} ^ {\prime}), \boldsymbol {z} _ {j} ^ {\prime}) - \nabla_ {\boldsymbol {w}} \ell (A _ {t} (S _ {1} ^ {\prime}), \boldsymbol {z} _ {j} ^ {\prime}) \right\| \\ + \left\| \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S ^ {\prime}\right), \boldsymbol {z} _ {i}\right) - \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {i}\right) \right\| \left\| \nabla_ {\boldsymbol {w}} \ell \left(A _ {t} \left(S _ {1} ^ {\prime}\right), \boldsymbol {z} _ {j} ^ {\prime}\right) \right\| d t \\ \leq \frac {1}{n} \sum_ {j = 2} ^ {n} \int_ {0} ^ {T} L _ {\ell} \beta_ {\ell} \frac {c t}{n} + \beta_ {\ell} \frac {c t}{n} L _ {\ell} d t \quad (\text {b y}) \\ = \left(1 - \frac {1}{n}\right) c L _ {\ell} \beta_ {\ell} \frac {T ^ {2}}{n} \\ \end{array}
$$