$$
\begin{array}{l} f (\bar {x} _ {T}) \\ \leq f (x _ {0}) - \frac {1}{8 \eta} \sum_ {t = 0} ^ {T - 1} \omega_ {t} - \frac {(1 - \lambda) ^ {2}}{2 5 6 \eta} \sum_ {t = 0} ^ {T - 1} \omega_ {t} - \frac {\eta}{2} \sum_ {t = 0} ^ {T - 1} \| \bar {v} _ {t} \| ^ {2} + (1 + \frac {7 6 8 \log (4 / \delta)}{n b _ {1} \beta}) \frac {L ^ {2} \eta}{n} \sum_ {t = 0} ^ {T - 1} \| X _ {t} - \bar {X} _ {t} \| _ {F} ^ {2} \\ + \sum_ {t = 0} ^ {T - 1} \frac {p _ {t} (\eta^ {2} C _ {v} ^ {2} + r ^ {2})}{\eta} + \frac {3 2 \log (4 / \delta) \beta \eta T \sigma^ {2}}{n b _ {1}} + (1 + \frac {3 8 4 \log (4 / \delta) L \eta}{n b _ {1} \beta}) \sum_ {t = 0} ^ {T - 1} L \omega_ {t} + \frac {4 \log (4 / \delta) \eta \sigma^ {2}}{n \beta b _ {0}} \\ \leq f \left(x _ {0}\right) - \frac {1}{8 \eta} \sum_ {t = 0} ^ {T - 1} \omega_ {t} - \frac {(1 - \lambda) ^ {2}}{2 5 6 \eta} \sum_ {t = 0} ^ {T - 1} \omega_ {t} - \frac {\eta}{2 n} \sum_ {t = 0} ^ {T - 1} \sum_ {i = 1} ^ {n} \left\| y _ {t} ^ {(i)} \right\| ^ {2} + \frac {\eta}{2 n} \sum_ {t = 0} ^ {T - 1} \left\| Y _ {t} - \bar {Y} _ {t} \right\| _ {F} ^ {2} \\ + \sum_ {t = 0} ^ {T - 1} \frac {p _ {t} \left(\eta^ {2} C _ {v} ^ {2} + r ^ {2}\right)}{\eta} + (1 + \frac {7 6 8 \log (4 / \delta)}{n b _ {1} \beta}) \frac {L ^ {2} \eta}{n} \sum_ {t = 0} ^ {T - 1} \| X _ {t} - \bar {X} _ {t} \| _ {F} ^ {2} + \frac {3 2 \log (4 / \delta) \beta \eta T \sigma^ {2}}{n b _ {1}} \\ + \left(1 + \frac {3 8 4 \log (4 / \delta) L \eta}{n b _ {1} \beta}\right) \sum_ {t = 0} ^ {T - 1} L \omega_ {t} + \frac {4 \log (4 / \delta) \eta \sigma^ {2}}{n \beta b _ {0}} \\ \leq f \left(x _ {0}\right) - \frac {1}{8 \eta} \sum_ {t = 0} ^ {T - 1} \omega_ {t} - \frac {(1 - \lambda) ^ {2}}{2 5 6 n \eta} \sum_ {t = 0} ^ {T - 1} \sum_ {i = 1} ^ {n} \left\| x _ {t + 1} ^ {(i)} - x _ {t} ^ {(i)} \right\| ^ {2} - \frac {\eta}{2 n} \sum_ {t = 0} ^ {T - 1} \sum_ {i = 1} ^ {n} \left\| y _ {t} ^ {(i)} \right\| ^ {2} \\ + \left(1 + \frac {7 6 8 \log (4 / \delta)}{n b _ {1} \beta} + \frac {(1 - \lambda) ^ {2}}{1 2 8 L ^ {2} \eta^ {2}}\right) \frac {L ^ {2} \eta}{n} \sum_ {t = 0} ^ {T - 1} \| X _ {t} - \bar {X} _ {t} \| _ {F} ^ {2} + \frac {\eta}{2 n} \sum_ {t = 0} ^ {T - 1} \| Y _ {t} - \bar {Y} _ {t} \| _ {F} ^ {2} \\ + \left(1 + \frac {3 8 4 \log (4 / \delta) L \eta}{n b _ {1} \beta}\right) \sum_ {t = 0} ^ {T - 1} L \omega_ {t} + \sum_ {t = 0} ^ {T - 1} \frac {p _ {t} \left(\eta^ {2} C _ {v} ^ {2} + r ^ {2}\right)}{\eta} + \frac {3 2 \log (4 / \delta) \beta \eta T \sigma^ {2}}{n b _ {1}} + \frac {4 \log (4 / \delta) \eta \sigma^ {2}}{n \beta b _ {0}} \\ \leq f \left(x _ {0}\right) - \frac {1}{8 L \eta} \sum_ {t = 0} ^ {T - 1} L \omega_ {t} - \frac {(1 - \lambda) ^ {2}}{2 5 6 n \eta} \sum_ {t = 0} ^ {T - 1} \sum_ {i = 1} ^ {n} \left\| x _ {t + 1} ^ {(i)} - x _ {t} ^ {(i)} \right\| ^ {2} - \frac {\eta}{2 n} \sum_ {t = 0} ^ {T - 1} \sum_ {i = 1} ^ {n} \left\| y _ {t} ^ {(i)} \right\| ^ {2} \\ + A _ {1} \sum_ {t = 0} ^ {T - 1} L \omega_ {t} + A _ {2} \frac {T \beta \eta \sigma^ {2}}{b _ {1}} + A _ {3} \frac {\eta \sigma^ {2}}{\beta b _ {0}} + A _ {4} \sum_ {t = 0} ^ {T - 1} \frac {p _ {t} \left(\eta^ {2} C _ {v} ^ {2} + r ^ {2}\right)}{\eta} + A _ {5} \frac {\eta}{n} \sum_ {i = 1} ^ {n} \| \nabla f _ {i} (x _ {0}) \| ^ {2} \tag {23} \\ \end{array}
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