$$
\begin{array}{l} \operatorname {E} \left[ f \left(x ^ {t + 1}\right) \right] + \frac {\gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \operatorname {E} \left[ \left\| g ^ {t + 1} - h ^ {t + 1} \right\| ^ {2} \right] + \frac {\gamma ((2 \omega + 1) p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}} \operatorname {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| g _ {i} ^ {t + 1} - h _ {i} ^ {t + 1} \right\| ^ {2} \right] \\ + \frac {\gamma}{b} \mathrm {E} \left[ \left\| h ^ {t + 1} - \nabla f (x ^ {t + 1}) \right\| ^ {2} \right] + \left(\frac {8 b \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma (p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}}\right) \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} [ f (x ^ {t}) ] - \frac {\gamma}{2} \operatorname {E} \left[ \| \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] \\ - \left(\frac {1}{2 \gamma} - \frac {L}{2} - \frac {1 2 \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} \left(\frac {2 (1 - b) ^ {2} L _ {\sigma} ^ {2}}{B} + 2 \widehat {L} ^ {2}\right) \right. \\ \left. - \frac {3 \gamma}{n p _ {\mathrm {a}} b} \left(\frac {2 (1 - b) ^ {2} L _ {\sigma} ^ {2}}{B} + 2 \left(1 - \frac {p _ {\mathrm {a a}}}{p _ {\mathrm {a}}}\right) \widehat {L} ^ {2}\right)\right) \mathrm {E} \left[ \| x ^ {t + 1} - x ^ {t} \| ^ {2} \right] \\ + \frac {\gamma}{b} \mathrm {E} \left[ \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] + \left(\frac {8 b \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma (p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}}\right) \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} (x ^ {t}) \right\| ^ {2} \right] \\ + \left(\frac {8 b ^ {2} \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma b}{n p _ {\mathrm {a}}} + \left(\frac {8 b \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma (p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}}\right) \frac {2 b ^ {2}}{p _ {\mathrm {a}}}\right) \frac {\sigma^ {2}}{B} \\ = \operatorname {E} [ f (x ^ {t}) ] - \frac {\gamma}{2} \operatorname {E} \left[ \left\| \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] \\ - \left(\frac {1}{2 \gamma} - \frac {L}{2} - \frac {2 4 \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} \left(\frac {(1 - b) ^ {2} L _ {\sigma} ^ {2}}{B} + \widehat {L} ^ {2}\right) \right. \\ \left. - \frac {6 \gamma}{n p _ {\mathrm {a}} b} \left(\frac {(1 - b) ^ {2} L _ {\sigma} ^ {2}}{B} + \left(1 - \frac {p _ {\mathrm {a a}}}{p _ {\mathrm {a}}}\right) \widehat {L} ^ {2}\right)\right) \mathrm {E} \left[ \left\| x ^ {t + 1} - x ^ {t} \right\| ^ {2} \right] \\ + \frac {\gamma}{b} \mathrm {E} \left[ \left\| h ^ {t} - \nabla f (x ^ {t}) \right\| ^ {2} \right] + \left(\frac {8 b \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma \left(p _ {\mathrm {a}} - p _ {\mathrm {a a}}\right)}{n p _ {\mathrm {a}} ^ {2}}\right) \mathrm {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} (x ^ {t}) \right\| ^ {2} \right] \\ + \left(\frac {8 b ^ {2} \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma b}{n p _ {\mathrm {a}}} + \left(\frac {8 b \gamma \omega (2 \omega + 1)}{n p _ {\mathrm {a}} ^ {2}} + \frac {2 \gamma (p _ {\mathrm {a}} - p _ {\mathrm {a a}})}{n p _ {\mathrm {a}} ^ {2}}\right) \frac {2 b ^ {2}}{p _ {\mathrm {a}}}\right) \frac {\sigma^ {2}}{B}. \\ \end{array}
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