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