$$
\begin{array}{l} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} _ {p} ^ {s, t} \right\| _ {F} ^ {2} \\ = \left\| \left[ \boldsymbol {Z} ^ {s, 0} - \frac {1}{\bar {P} L _ {Z} ^ {s}} \sum_ {j = 0} ^ {t - 1} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right] - \left[ \boldsymbol {Z} _ {p} ^ {s, 0} - \frac {1}{L _ {Z} ^ {s}} \sum_ {j = 0} ^ {t - 1} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right] \right\| _ {F} ^ {2} \\ = \frac {1}{\left(L _ {Z} ^ {s}\right) ^ {2}} \left\| \frac {1}{\bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) - \sum_ {j = 0} ^ {t - 1} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ \leq \frac {t}{\left(L _ {Z} ^ {s}\right) ^ {2}} \sum_ {j = 0} ^ {t - 1} \left\| \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ = \frac {t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P} ^ {2}} \sum_ {j = 0} ^ {t - 1} \left\| \sum_ {p ^ {\prime} \in \mathcal {A} ^ {s}} \left[ \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, j}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right] \right\| _ {F} ^ {2} \tag {64} \\ \leq \frac {t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p ^ {\prime} \in \mathcal {A} ^ {s}} \left\| \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, j}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ = \frac {t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \underbrace {\left\| \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s , j} , \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s , j} , \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2}} _ {T. 6} \\ \leq \frac {t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \underbrace {\left[ 4 \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s , j} - \boldsymbol {Z} _ {p} ^ {s , j} \right\| _ {F} ^ {2} + 4 \left(L _ {Z _ {p ^ {\prime}}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s , j} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s , j} \right\| _ {F} ^ {2} + 8 \zeta^ {2} \right]} _ {\text {B o u n d o f T . 6}} \\ = \frac {4 t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \frac {8 t ^ {2} \zeta^ {2}}{\left(L _ {Z} ^ {s}\right) ^ {2} m} \\ + \frac {4 t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \left(L _ {Z _ {p ^ {\prime}}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s, j} \right\| _ {F} ^ {2} \\ \end{array}
$$