$$
\begin{array}{l} \mathrm {E} \left[ f (x ^ {t + 1}) \right] + \frac {\gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \| g ^ {t + 1} - h ^ {t + 1} \| ^ {2} \right] + \frac {\gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \| g _ {i} ^ {t + 1} - h _ {i} ^ {t + 1} \| ^ {2} \right] \\ + \nu \mathrm {E} \left[ \left\| h ^ {t + 1} - \nabla f (x ^ {t + 1}) \right\| ^ {2} \right] + \rho \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t + 1} - \nabla f _ {i} (x ^ {t + 1}) \right\| ^ {2} \right] \\ \leq \operatorname {E} \left[ f (x ^ {t}) - \frac {\gamma}{2} \left\| \nabla f (x ^ {t}) \right\| ^ {2} - \left(\frac {1}{2 \gamma} - \frac {L}{2}\right) \left\| x ^ {t + 1} - x ^ {t} \right\| ^ {2} + \gamma \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] \\ + \frac {\gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \left\| g ^ {t} - h ^ {t} \right\| ^ {2} \right] + \frac {\gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| g _ {i} ^ {t} - h _ {i} ^ {t} \right\| ^ {2} \right] \\ + \frac {4 \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| k _ {i} ^ {t + 1} \right\| ^ {2} \right] \\ + \nu \mathrm {E} \left[ \left\| h ^ {t + 1} - \nabla f (x ^ {t + 1}) \right\| ^ {2} \right] + \rho \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t + 1} - \nabla f _ {i} (x ^ {t + 1}) \right\| ^ {2} \right] \\ = \operatorname {E} \left[ f (x ^ {t}) - \frac {\gamma}{2} \left\| \nabla f (x ^ {t}) \right\| ^ {2} - \left(\frac {1}{2 \gamma} - \frac {L}{2}\right) \left\| x ^ {t + 1} - x ^ {t} \right\| ^ {2} + \gamma \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] \\ + \frac {\gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \left\| g ^ {t} - h ^ {t} \right\| ^ {2} \right] + \frac {\gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| g _ {i} ^ {t} - h _ {i} ^ {t} \right\| ^ {2} \right] \\ + \frac {4 \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \mathrm {E} _ {k} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \| k _ {i} ^ {t + 1} \| ^ {2} \right] \right] \\ \left. \right. + \nu \mathrm {E} \left[ \mathrm {E} _ {B} \left[ \mathrm {E} _ {p _ {\mathrm {a}}} \left[\left\| h ^ {t + 1} - \nabla f \left(x ^ {t + 1}\right)\right\| ^ {2} \right]\right]\right] \\ \left. \right. + \rho \mathrm {E} \left[ \right. \mathrm {E} _ {B} \left[ \right. \mathrm {E} _ {p _ {\mathrm {a}}} \left[ \right. \frac {1}{n} \sum_ {i = 1} ^ {n} \left| \right.\left| h _ {i} ^ {t + 1} - \nabla f _ {i} \left(x ^ {t + 1}\right)\right| ^ {2} \left. \right]\left. \right]\left. \right] \\ \leq \operatorname {E} \left[ f (x ^ {t}) - \frac {\gamma}{2} \| \nabla f (x ^ {t}) \| ^ {2} - \left(\frac {1}{2 \gamma} - \frac {L}{2}\right) \| x ^ {t + 1} - x ^ {t} \| ^ {2} + \gamma \| h ^ {t} - \nabla f (x ^ {t}) \| ^ {2} \right] \\ + \frac {\gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \left\| g ^ {t} - h ^ {t} \right\| ^ {2} \right] + \frac {\gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| g _ {i} ^ {t} - h _ {i} ^ {t} \right\| ^ {2} \right] \\ + \frac {4 \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \frac {2 b ^ {2} \sigma^ {2}}{B} + \left(\frac {2 (1 - b) ^ {2} L _ {\sigma} ^ {2}}{B} + 2 \widehat {L} ^ {2}\right) \left\| x ^ {t + 1} - x ^ {t} \right\| ^ {2} + 2 b ^ {2} \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} (x ^ {t}) \right\| ^ {2} \right] \\ \end{array}
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