SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| \nabla_ {Z} F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \tag {52} \\ \leq \frac {1 6 \zeta^ {2}}{\bar {P}} + 2 (1 + \frac {8}{\bar {P}}) \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \\ \end{array}
$$
Proof. For any  $s$  and  $t$ , it holds that
$$
\begin{array}{l} \left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \right\| _ {F} ^ {2} \\ = \left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}) - \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \right. \\ + \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \| _ {F} ^ {2} \tag {53} \\ \leq 2 \underbrace {\left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s , t - 1} , \boldsymbol {C} ^ {s}) - \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s , t - 1} , \boldsymbol {C} _ {p} ^ {s}) \right\| _ {F} ^ {2}} _ {\text {T . 4}} \\ + 2 \underbrace {\left\| \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s , t - 1} , \boldsymbol {C} _ {p} ^ {s}\right) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s , t - 1} , \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2}} _ {\text {T . 5}} \\ \end{array}
$$
where we used the fact that  $\| \sum_{i = 1}^{n}a_{i}\|_{2}^{2}\leq n\sum_{i = 1}^{n}\| a_{i}\|_{2}^{2}$
Firstly, we give a bound on T.4:
$$
\begin{array}{l} \left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}) - \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \right\| _ {F} ^ {2} \\ = \left\| \sum_ {p = 1} ^ {P} \omega_ {p} \left[ \nabla_ {Z} f \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right] \right\| _ {F} ^ {2} \tag {54} \\ \leq \sum_ {p = 1} ^ {P} \omega_ {p} \left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) - \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \right\| _ {F} ^ {2} \\ \leq \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \\ \end{array}
$$
where we used the fact that  $\nabla_Z f_p(\cdot, C_p^s)$  is  $L_{Z_p}^s$ -Lipschitz continuous.
Secondly, we give a bound on T.5:
$$
\begin{array}{l} \left\| \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ = \frac {1}{\bar {P} ^ {2}} \left\| \sum_ {p ^ {\prime} \in \mathcal {A} ^ {s}} \left[ \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right] \right\| _ {F} ^ {2} \\ \leq \frac {1}{\bar {P}} \sum_ {p ^ {\prime} \in \mathcal {A} ^ {s}} \left\| \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right\| _ {F} ^ {2} \tag {55} \\ = \frac {1}{\bar {\bar {P}}} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \left\| \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s, t - 1}, \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right\| _ {F} ^ {2} \\ \leq \frac {1}{\bar {P}} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \sum_ {p = 1} ^ {P} \omega_ {p} \underbrace {\left\| \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s , t - 1} , \boldsymbol {C} _ {p} ^ {s}\right) - \nabla_ {Z} f _ {p ^ {\prime}} \left(\boldsymbol {Z} _ {p ^ {\prime}} ^ {s , t - 1} , \boldsymbol {C} _ {p ^ {\prime}} ^ {s}\right) \right\| _ {F} ^ {2}} _ {\text {T . 6}} \\ \end{array}
$$
Thirdly, we give a bound on T.6:
Here, it is the second time that we use the facts that  $\| \sum_{i=1}^{n} a_i\|_2^2 \leq n \sum_{i=1}^{n} \|a_i\|_2^2$  and that  $\nabla_Z f_p(\cdot, C_p^s)$  is  $L_{Z_p}^s$ -Lipschitz continuous.
Based on the bound of T.6, we derive the bound on  $T.5$ .
$$
\begin{array}{l} \left\| \sum_ {p = 1} ^ {P} \omega_ {p} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \right\| _ {F} ^ {2} \\ \leq \frac {1}{\bar {P}} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \sum_ {p = 1} ^ {P} \omega_ {p} \underbrace {\left[ 4 (L _ {Z _ {p}} ^ {s}) ^ {2} \left\| \boldsymbol {Z} ^ {s , t - 1} - \boldsymbol {Z} _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2} + 4 (L _ {Z _ {p ^ {\prime}}} ^ {s}) ^ {2} \left\| \boldsymbol {Z} ^ {s , t - 1} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s , t - 1} \right\| _ {F} ^ {2} + 8 \zeta^ {2} \right]} _ {\text {B o u n d o f T . 5}} \\ \leq \frac {8 \zeta^ {2}}{\bar {P}} + \frac {8}{\bar {P}} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \tag {57} \\ \end{array}
$$
Fourthly, based on the bounds of  $T.4$  and  $T.5$ , we continue to derive the final required bound.
$$
\begin{array}{l} \left\| \nabla_ {Z} F (\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}) - \frac {1}{\bar {P}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \right\| _ {F} ^ {2} \\ \leq 2 \underbrace {\sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| Z ^ {s , t - 1} - Z _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2}} _ {\text {B o u n d o f T . 3}} + 2 \underbrace {\left[ \frac {8 \zeta^ {2}}{\bar {P}} + \frac {8}{\bar {P}} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| Z ^ {s , t - 1} - Z _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2} \right]} _ {\text {B o u n d o f T . 4}} \\ = \frac {1 6 \zeta^ {2}}{\bar {P}} + 2 \left(1 + \frac {8}{\bar {P}}\right) \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \tag {58} \\ \end{array}
$$
Based on Lemma 2, we continue to do the following derivation.
$$
\begin{array}{l} F \left(\boldsymbol {Z} ^ {s, t}, \boldsymbol {C} ^ {s}\right) - F \left(\boldsymbol {Z} ^ {s, t - 1}, \boldsymbol {C} ^ {s}\right) \\ \leq - \frac {L _ {Z} ^ {s}}{4} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} ^ {s, t - 1} \right\| _ {F} ^ {2} \\ + \frac {1}{L _ {Z} ^ {s}} \underbrace {\left[ \frac {1 6 \zeta^ {2}}{\bar {P}} + 2 \left(1 + \frac {8}{\bar {P}}\right) \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| Z ^ {s , t - 1} - Z _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2} \right]} _ {\text {B o u n d o f T . 3}} \tag {59} \\ \end{array}
$$
$$
\begin{array}{l} = - \frac {L _ {Z} ^ {s}}{4} \left\| Z ^ {s, t} - Z ^ {s, t - 1} \right\| _ {F} ^ {2} + \frac {1 6 \zeta^ {2}}{\bar {P} L _ {Z} ^ {s}} \\ + \frac {2}{L _ {Z} ^ {s}} \left(1 + \frac {8}{\bar {P}}\right) \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \\ \end{array}
$$
Then, summing it up from  $t = 1$  to  $Q$  yields
$$
\begin{array}{l} F \left(\boldsymbol {Z} ^ {s, Q}, \boldsymbol {C} ^ {s}\right) - F \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s}\right) \\ \leq - \frac {L _ {Z} ^ {s}}{4} \sum_ {t = 1} ^ {Q} \| Z ^ {s, t} - Z ^ {s, t - 1} \| _ {F} ^ {2} + \frac {1 6 Q \zeta^ {2}}{\bar {P} L _ {Z} ^ {s}} \tag {60} \\ + \frac {2}{L _ {Z} ^ {s}} (1 + \frac {8}{\bar {P}}) \underbrace {\sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} (L _ {Z _ {p}} ^ {s}) ^ {2} \left\| \boldsymbol {Z} ^ {s , t - 1} - \boldsymbol {Z} _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2}} _ {\text {T . 7}} \\ \end{array}
$$
Now, we give a bound on T.7.
$$
\begin{array}{l} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} _ {p} ^ {s, t} \right\| _ {F} ^ {2} \\ \leq \frac {4 t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \frac {8 t ^ {2} \zeta^ {2}}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \tag {61} \\ + \frac {4 t}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 1} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \left(L _ {Z _ {p ^ {\prime}}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s, j} \right\| _ {F} ^ {2} \\ \end{array}
$$
Proof. Based on the update schemes of  $C$  and  $Z$ , we have
$$
\begin{array}{l} \boldsymbol {Z} _ {p} ^ {s, t} = \boldsymbol {Z} _ {p} ^ {s, t - 1} - \frac {1}{L _ {Z} ^ {s}} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}) \\ \Longleftrightarrow \boldsymbol {Z} ^ {s, t} = \boldsymbol {Z} ^ {s, t - 1} - \frac {1}{\bar {P} L _ {Z} ^ {s}} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, t - 1}, \boldsymbol {C} _ {p} ^ {s}\right) \tag {62} \\ \end{array}
$$
$$
\begin{array}{l} \boldsymbol {Z} _ {p} ^ {s, t} = \boldsymbol {Z} _ {p} ^ {s, 0} - \frac {1}{L _ {Z} ^ {s}} \sum_ {j = 0} ^ {t - 1} \nabla_ {Z} f _ {p} (\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}) \\ \Longleftrightarrow \boldsymbol {Z} ^ {s, t} = \boldsymbol {Z} ^ {s, 0} - \frac {1}{\bar {P} L _ {Z} ^ {s}} \sum_ {j = 0} ^ {t - 1} \sum_ {p \in \mathcal {A} ^ {s}} \nabla_ {Z} f _ {p} \left(\boldsymbol {Z} _ {p} ^ {s, j}, \boldsymbol {C} _ {p} ^ {s}\right) \tag {63} \\ \end{array}
$$
where  $Z_{p}^{s,0} = Z^{s,0} = Z^{s - 1}$
Lemma E.3. For any  $s$ , it holds that
Based on the above identities,
Based on Lemma 3, we give a bound on T.7.
Lemma E.4. For any  $s$ , it holds that
$$
\sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \leq \frac {8 Q (Q - 1) (2 Q - 1) \zeta^ {2}}{\bar {P} - 4 (Q - 1) ^ {2} \left(1 + \bar {L} _ {Z} ^ {2} / \underline {{L}} _ {Z} ^ {2}\right)} \tag {65}
$$
$$
\begin{array}{l} \sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} (L _ {Z _ {p}} ^ {s}) ^ {2} \left| \left| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right| \right| _ {F} ^ {2} \\ \leq \sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left[ \frac {4 (t - 1)}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 2} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \right] \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \frac {8 (t - 1) ^ {2} \zeta^ {2}}{\left(L _ {Z} ^ {s}\right) ^ {2} \bar {P}} \\ \left. + \frac {4 (t - 1)}{(L _ {Z} ^ {s}) ^ {2} \bar {P}} \sum_ {j = 0} ^ {t - 2} \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} \right] \\ = \sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left(\frac {L _ {Z _ {p}} ^ {s}}{L _ {Z} ^ {s}}\right) ^ {2} \sum_ {j = 0} ^ {t - 2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \sum_ {t = 1} ^ {Q} \frac {8 (t - 1) ^ {2} \zeta^ {2}}{\bar {P}} \\ + \sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \sum_ {j = 0} ^ {t - 2} \sum_ {p ^ {\prime} = 1} ^ {P} \omega_ {p ^ {\prime}} \left(L _ {Z _ {p ^ {\prime}}} ^ {s}\right) ^ {2} \| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p ^ {\prime}} ^ {s, j} \| _ {F} ^ {2} \tag {66} \\ \leq \sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left(\frac {\bar {L} _ {Z}}{\underline {{L}} _ {Z}}\right) ^ {2} \sum_ {j = 0} ^ {t - 2} \left\| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \sum_ {t = 1} ^ {Q} \frac {8 (t - 1) ^ {2} \zeta^ {2}}{\bar {P}} \\ + \sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \sum_ {j = 0} ^ {t - 2} \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} \\ = \sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \left(1 + \frac {\bar {L} _ {Z} ^ {2}}{\underline {{L}} _ {Z} ^ {2}}\right) \sum_ {j = 0} ^ {t - 2} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| Z ^ {s, j} - Z _ {p} ^ {s, j} \right\| _ {F} ^ {2} + \frac {8 Q (Q - 1) (2 Q - 1) \zeta^ {2}}{\bar {P}} \\ \leq \frac {4 (Q - 1) ^ {2}}{\bar {P}} (1 + \frac {\overline {{L}} _ {Z} ^ {2}}{\underline {{L}} _ {Z} ^ {2}}) \sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} (L _ {Z _ {p}} ^ {s}) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} + \frac {8 Q (Q - 1) (2 Q - 1) \zeta^ {2}}{\bar {P}} \\ \end{array}
$$
Here, we used the inequality as
$$
\sum_ {t = 1} ^ {Q} \frac {4 (t - 1)}{\bar {P}} \sum_ {j = 0} ^ {t - 2} \| \boldsymbol {Z} ^ {s, j} - \boldsymbol {Z} _ {p} ^ {s, j} \| _ {F} ^ {2} \leq \frac {4 (Q - 1) ^ {2}}{\bar {P}} \sum_ {t = 1} ^ {Q} \| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \| _ {F} ^ {2} \tag {67}
$$
$$
\sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} \left(L _ {Z _ {p}} ^ {s}\right) ^ {2} \left\| \boldsymbol {Z} ^ {s, t - 1} - \boldsymbol {Z} _ {p} ^ {s, t - 1} \right\| _ {F} ^ {2} \leq \frac {8 Q (Q - 1) (2 Q - 1) \zeta^ {2}}{\bar {P} - 4 (Q - 1) ^ {2} \left(1 + \bar {L} _ {Z} ^ {2} / \underline {{L}} _ {Z} ^ {2}\right)} \tag {68}
$$
Proof. Based on Lemma 3, we have
Based on Lemma 4, we continue to do the following derivation:
$$
\begin{array}{l} F \left(\boldsymbol {Z} ^ {s, Q}, \boldsymbol {C} ^ {s}\right) - F \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s}\right) \\ \leq - \frac {L _ {Z} ^ {s}}{4} \sum_ {t = 1} ^ {Q} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} ^ {s, t - 1} \right\| _ {F} ^ {2} + \frac {1 6 Q \zeta^ {2}}{\bar {P} L _ {Z} ^ {s}} \\ + \frac {2}{L _ {Z} ^ {s}} (1 + \frac {8}{\bar {P}}) \underbrace {\sum_ {t = 1} ^ {Q} \sum_ {p = 1} ^ {P} \omega_ {p} (L _ {Z _ {p}} ^ {s}) ^ {2} \left\| Z ^ {s , t - 1} - Z _ {p} ^ {s , t - 1} \right\| _ {F} ^ {2}} _ {T. 7} \\ \leq - \frac {L _ {Z} ^ {s}}{4} \sum_ {t = 1} ^ {Q} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} ^ {s, t - 1} \right\| _ {F} ^ {2} + \frac {1 6 Q \zeta^ {2}}{\bar {P} L _ {Z} ^ {s}} \tag {69} \\ + \frac {2}{L _ {Z} ^ {s}} (1 + \frac {8}{\bar {P}}) \underbrace {\frac {8 Q (Q - 1) (2 Q - 1) \zeta^ {2}}{\bar {P} - 4 (Q - 1) ^ {2} \left(1 + \overline {{L}} _ {Z} ^ {2} / \underline {{L}} _ {Z} ^ {2}\right)}} _ {B o u n d o f T. 7} \\ \leq - \frac {L _ {Z} ^ {s}}{4} \sum_ {t = 1} ^ {Q} \left\| \boldsymbol {Z} ^ {s, t} - \boldsymbol {Z} ^ {s, t - 1} \right\| _ {F} ^ {2} + \frac {1 6 Q \zeta^ {2} \psi}{\bar {P} L _ {Z} ^ {s}} \\ \end{array}
$$
where  $\psi = 1 + \frac{(\bar{P} + 8)(Q - 1)(2Q - 1)}{\bar{P} - 4(Q - 1)^2(1 + \overline{L}_Z^2 / \underline{L}_Z^2)}$ .
Then, combining (46) and (69) yields
$$
\begin{array}{l} \frac {1}{2} \left(\gamma_ {\text {m i n}} ^ {s} + \lambda\right) \sum_ {p = 1} ^ {P} \omega_ {p} \left\| C _ {p} ^ {s} - C _ {p} ^ {s - 1} \right\| _ {F} ^ {2} + \frac {L _ {Z} ^ {s}}{4} \sum_ {t = 1} ^ {Q} \left\| Z ^ {s, t} - Z ^ {s, t - 1} \right\| _ {F} ^ {2} \tag {70} \\ \leq \left[ F \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s - 1}\right) - F \left(\boldsymbol {Z} ^ {s, Q}, \boldsymbol {C} ^ {s}\right) \right] + \frac {1 6 Q \zeta^ {2} \psi}{\bar {P} L _ {Z} ^ {s}} \\ \end{array}
$$
Derivation of the main result: Based on (70), we derive the convergence in terms of the iterative terms,  $T_{C}(Z^{s,0}, C^{s})$  and  $T_{Z}(Z^{s,t}, C^{s})$  for  $s = 1, 2, \ldots, S$ .
$$
T _ {C} \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s}\right) \leq \frac {2}{\gamma_ {\text {m i n}} ^ {s} + \lambda} \left[ F \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s - 1}\right) - F \left(\boldsymbol {Z} ^ {s, Q}, \boldsymbol {C} ^ {s}\right) \right] + \frac {3 2 Q \zeta^ {2} \psi}{\left(\gamma_ {\text {m i n}} ^ {s} + \lambda\right) \bar {P} L _ {Z} ^ {s}} \tag {71}
$$
Then, summing it up from  $s = 1$  to  $S$  yields
$$
\sum_ {s = 1} ^ {S} T _ {C} \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s}\right) \leq \frac {2}{\underline {{\gamma}} _ {\text {m i n}} + \lambda} \left[ F \left(\boldsymbol {Z} ^ {1, 0}, \boldsymbol {C} ^ {0}\right) - \underline {{f}} \right] + \frac {3 2 S Q \zeta^ {2} \psi}{\left(\underline {{\gamma}} _ {\text {m i n}} + \lambda\right) \bar {P} \underline {{L}} _ {Z}} \tag {72}
$$
$$
\sum_ {t = 1} ^ {Q} T _ {Z} \left(\boldsymbol {Z} ^ {s, t}, \boldsymbol {C} ^ {s}\right) \leq \frac {4}{L _ {Z} ^ {s}} \left[ F \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s - 1}\right) - F \left(\boldsymbol {Z} ^ {s, Q}, \boldsymbol {C} ^ {s}\right) \right] + \frac {6 4 Q \zeta^ {2} \psi}{\bar {P} \left(L _ {Z} ^ {s}\right) ^ {2}} \tag {73}
$$
$$
\sum_ {s = 1} ^ {S} \sum_ {t = 1} ^ {Q} T _ {Z} \left(\boldsymbol {Z} ^ {s, t}, \boldsymbol {C} ^ {s}\right) \leq \frac {4}{\underline {{L}} _ {Z}} \left[ F \left(\boldsymbol {Z} ^ {1, 0}, \boldsymbol {C} ^ {0}\right) - \underline {{f}} \right] + \frac {6 4 S Q \zeta^ {2} \psi}{\bar {P} \underline {{L}} _ {Z} ^ {2}} \tag {74}
$$
Lastly, combining (72) and (74) and dividing two sides of it by  $T = S(1 + Q)$  yields
$$
\frac {1}{T} \left[ \sum_ {s = 1} ^ {S} T _ {C} \left(\boldsymbol {Z} ^ {s, 0}, \boldsymbol {C} ^ {s}\right) + \sum_ {s = 1} ^ {S} \sum_ {t = 1} ^ {Q} T _ {Z} \left(\boldsymbol {Z} ^ {s, t}, \boldsymbol {C} ^ {s}\right) \right] \leq \frac {D}{T} \left[ F \left(\boldsymbol {Z} ^ {1, 0}, \boldsymbol {C} ^ {0}\right) - \underline {{f}} \right] + \frac {1 6 \zeta^ {2} \psi D}{\bar {P} \underline {{L}} _ {Z}} \tag {75}
$$
where  $D = \frac{2}{\underline{\gamma}_{min} + \lambda} +\frac{4}{L_Z}$
F Proof for theorem on error bound on noisy similarity matrix
Proof. It follows from the triangle inequality of matrix norm that
$$
\begin{array}{l} \left\| \hat {\boldsymbol {K}} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {x x} \right\| _ {\infty} = \left\| \hat {\boldsymbol {K}} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {\tilde {x} \tilde {x}} + \boldsymbol {K} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {x x} \right\| _ {\infty} \tag {76} \\ \leq \left\| \hat {\boldsymbol {K}} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {\tilde {x} \tilde {x}} \right\| _ {\infty} + \| \boldsymbol {K} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {x x} \| _ {\infty}. \\ \end{array}
$$
Since  $\tilde{\pmb{X}} = \pmb {X} + \pmb{E}$  and  $\phi (\tilde{\pmb{X}}) = \phi (\pmb {Z})\pmb{C}$ , we have
$$
\left\{ \begin{array}{l} \hat {\boldsymbol {K}} _ {\tilde {x} \tilde {x}} = (\phi (\boldsymbol {Z}) \boldsymbol {C}) ^ {T} (\phi (\boldsymbol {Z}) \boldsymbol {C}) = \boldsymbol {C} ^ {T} \mathcal {K} (\boldsymbol {Z}, \boldsymbol {Z}) \boldsymbol {C} \\ \boldsymbol {K} _ {\tilde {x} \tilde {x}} = \phi (\tilde {\boldsymbol {X}}) ^ {T} \phi (\tilde {\boldsymbol {X}}) = \mathcal {K} (\tilde {\boldsymbol {X}}, \tilde {\boldsymbol {X}}) \\ \boldsymbol {K} _ {x x} = \phi (\boldsymbol {X}) ^ {T} \phi (\boldsymbol {X}) = \mathcal {K} (\boldsymbol {X}, \boldsymbol {X}) \end{array} \right. \tag {77}
$$
where  $\pmb{E}$  is a Gaussian noise matrix, of which  $E_{i,j} \sim \mathcal{N}(0,\sigma^2)$ . Hence, we obtain
$$
\left\| \hat {\boldsymbol {K}} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {x x} \right\| _ {\infty} \leq \left\| \boldsymbol {C} ^ {T} \mathcal {K} (\boldsymbol {Z}, \boldsymbol {Z}) \boldsymbol {C} - \mathcal {K} (\tilde {\boldsymbol {X}}, \tilde {\boldsymbol {X}}) \right\| _ {\infty} + \left\| \mathcal {K} (\tilde {\boldsymbol {X}}, \tilde {\boldsymbol {X}}) - \mathcal {K} (\boldsymbol {X}, \boldsymbol {X}) \right\| _ {\infty}. \tag {78}
$$
Denote by  $\pmb{x}_i$  (or  $e_i$ ) the  $i$ -th column of  $\pmb{X} \in \mathbb{R}^{m \times n}$  (or  $\pmb{E} \in \mathbb{R}^{m \times n}$ ),  $i = 1, \dots, n$ , we first derive the upper bound of the second term of the RHS of (78) as follows:
$$
\begin{array}{l} \left\| \boldsymbol {K} _ {\tilde {x} \tilde {x}} - \boldsymbol {K} _ {x x} \right\| _ {\infty} = \left\| \mathcal {K} (\tilde {\boldsymbol {X}}, \tilde {\boldsymbol {X}}) - \mathcal {K} (\boldsymbol {X}, \boldsymbol {X}) \right\| _ {\infty} \\ = \max  _ {i, j} \left| \mathcal {K} \left(\tilde {\boldsymbol {x}} _ {i}, \tilde {\boldsymbol {x}} _ {j}\right) - \mathcal {K} \left(\boldsymbol {x} _ {i}, \boldsymbol {x} _ {j}\right) \right| \\ = \max  _ {i, j} \left| \exp \left(- \frac {\left\| \left(\boldsymbol {x} _ {i} + \boldsymbol {e} _ {i}\right) - \left(\boldsymbol {x} _ {j} + \boldsymbol {e} _ {j}\right) \right\| _ {2} ^ {2}}{2 r ^ {2}}\right) - \exp \left(- \frac {\left\| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \right\| _ {2} ^ {2}}{2 r ^ {2}}\right) \right| \\ \leq \max  _ {i, j} \frac {1}{2 r ^ {2}} \left| - \| \left(\boldsymbol {x} _ {i} + \boldsymbol {e} _ {i}\right) - \left(\boldsymbol {x} _ {j} + \boldsymbol {e} _ {j}\right) \| _ {2} ^ {2} + \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \| _ {2} ^ {2} \right| \tag {79} \\ = \max _ {i, j} \frac {1}{2 r ^ {2}} \left| \left\| (\boldsymbol {x} _ {i} - \boldsymbol {x} _ {j}) + (\boldsymbol {e} _ {i} - \boldsymbol {e} _ {j}) \right\| _ {2} ^ {2} - \left\| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \right\| _ {2} ^ {2} \right| \\ = \max  _ {i, j} \frac {1}{2 r ^ {2}} \left| \| \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \| _ {2} ^ {2} + 2 \langle \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j}, \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \rangle \right| \\ \leq \max  _ {i, j} \frac {1}{2 r ^ {2}} \left(\| \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \| _ {2} ^ {2} + 2 | \langle \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j}, \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \rangle |\right) \\ \leq \max _ {i, j} \frac {1}{2 r ^ {2}} \left(\| \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \| _ {2} ^ {2} + 2 \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \| _ {2} \| \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \| _ {2}\right), \\ \end{array}
$$
where for the first inequality we have used the fact that the exponential function is locally Lipschitz continuous, i.e.,
$$
\left| e ^ {x} - e ^ {y} \right| <   \left| x - y \right| \text {f o r} x, y <   0.
$$
Now, let us figure out the upper bound of  $\| \pmb{e}_i - \pmb{e}_j\| _2^2$ . Note that
$$
\left\| \boldsymbol {e} _ {i} - \boldsymbol {e} _ {j} \right\| _ {2} ^ {2} = \sum_ {l = 1} ^ {m} \left(e _ {l i} - e _ {l j}\right) ^ {2} = 2 \sigma^ {2} \sum_ {l = 1} ^ {m} \left(\frac {e _ {l i} - e _ {l j}}{\sqrt {2} \sigma}\right) ^ {2} \tag {80}
$$
where  $e_{li}$  represents the  $l$ -th element of the column vector  $\boldsymbol{e}_i$ ,  $k = 1, \dots, m$ . It is clear that for  $l = 1, \dots, m$ ,