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