$$
\begin{array}{l} \leq \operatorname {E} [ f (x ^ {t}) ] - \frac {\gamma}{2} \operatorname {E} \left[ \| \nabla f (x ^ {t}) \| ^ {2} \right] \\ - \left(\frac {1}{2 \gamma} - \frac {L}{2} - \frac {2 \gamma (2 \omega + 1) \omega}{n p _ {\mathrm {a}} ^ {2}} \left(\frac {L _ {\sigma} ^ {2}}{B} + \widehat {L} ^ {2}\right) - \frac {2 \gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}}) \omega}{n p _ {\mathrm {a}} ^ {3}} \left(\frac {L _ {\sigma} ^ {2}}{B} + \widehat {L} ^ {2}\right) \right. \\ - \nu \left(\frac {2 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) - \rho \left(\frac {2 L _ {\sigma} ^ {2}}{p _ {\mathrm {a}} B} + \frac {2 (1 - p _ {\mathrm {a}}) \widehat {L} ^ {2}}{p _ {\mathrm {a}}}\right) \Bigg) \mathrm {E} \left[ \| x ^ {t + 1} - 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] \\ + (\gamma + \nu (1 - b)) E \left[ \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] \\ + \left(\nu \frac {2 \left(p _ {\mathrm {a}} - p _ {\mathrm {a a}}\right) b ^ {2}}{n p _ {\mathrm {a}} ^ {2} p _ {\mathrm {m e g a}}} + \rho (1 - b)\right) \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left| \left| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right| \right| ^ {2} \right] \\ + \left(\frac {2 \nu b ^ {2}}{n p _ {\mathrm {m e g a}} p _ {\mathrm {a}}} + \frac {2 \rho b ^ {2}}{p _ {\mathrm {a}} p _ {\mathrm {m e g a}}}\right) \frac {\sigma^ {2}}{B ^ {\prime}} \\ \leq \operatorname {E} [ f (x ^ {t}) ] - \frac {\gamma}{2} \operatorname {E} \left[ \| \nabla f (x ^ {t}) \| ^ {2} \right] \\ - \left(\frac {1}{2 \gamma} - \frac {L}{2} - \frac {4 \gamma (2 \omega + 1) \omega}{n p _ {\mathrm {a}} ^ {2}} \left(\frac {L _ {\sigma} ^ {2}}{B} + \hat {L} ^ {2}\right) \right. \\ \left. - \nu \left(\frac {2 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) - \rho \left(\frac {2 L _ {\sigma} ^ {2}}{p _ {\mathrm {a}} B} + \frac {2 (1 - p _ {\mathrm {a}}) \widehat {L} ^ {2}}{p _ {\mathrm {a}}}\right)\right) \mathrm {E} \left[ \| x ^ {t + 1} - 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] \\ + (\gamma + \nu (1 - b)) E \left[ \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] \\ + \left(\nu \frac {2 \left(p _ {\mathrm {a}} - p _ {\mathrm {a a}}\right) b ^ {2}}{n p _ {\mathrm {a}} ^ {2} p _ {\mathrm {m e g a}}} + \rho (1 - b)\right) \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left| \left| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right| \right| ^ {2} \right] \\ + \left(\frac {2 \nu b ^ {2}}{n p _ {\text {m e g a}} p _ {\mathrm {a}}} + \frac {2 \rho b ^ {2}}{p _ {\mathrm {a}} p _ {\text {m e g a}}}\right) \frac {\sigma^ {2}}{B ^ {\prime}}, \\ \end{array}
$$