$$
\begin{array}{l} \operatorname {E} \left[ f (x ^ {t + 1}) \right] + \frac {2 \gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \operatorname {E} \left[ \left\| g ^ {t + 1} - h ^ {t + 1} \right\| ^ {2} \right] + \frac {4 \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] \\ + \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 \| h ^ {t} - \nabla f (x ^ {t}) \| ^ {2} \right] \\ + (1 - \gamma \mu) \frac {2 \gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \left\| g ^ {t} - h ^ {t} \right\| ^ {2} \right] + (1 - \gamma \mu) \frac {4 \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 {1 0 \gamma (2 \omega + 1) \omega}{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] \\ \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] \\ + (1 - \gamma \mu) \frac {2 \gamma (2 \omega + 1)}{p _ {\mathrm {a}}} \mathrm {E} \left[ \| g ^ {t} - h ^ {t} \| ^ {2} \right] + (1 - \gamma \mu) \frac {4 \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} - h _ {i} ^ {t} \| ^ {2} \right] \\ + \frac {1 0 \gamma (2 \omega + 1) \omega}{n p _ {\mathrm {a}} ^ {2}} \mathrm {E} \left[ \left(2 \widehat {L} ^ {2} + \frac {(1 - p _ {\text {p a g e}}) L _ {\max} ^ {2}}{B}\right) \left\| x ^ {t + 1} - x ^ {t} \right\| ^ {2} + \frac {2 b ^ {2}}{p _ {\text {p a g e}}} \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \right] \\ + \nu \mathrm {E} \left(\left(\frac {2 \left(p _ {\mathrm {a}} - p _ {\mathrm {a a}}\right) \widehat {L} ^ {2}}{n p _ {\mathrm {a}} ^ {2}} + \frac {(1 - p _ {\text {p a g e}}) L _ {\max } ^ {2}}{n p _ {\mathrm {a}} B}\right) \| x ^ {t + 1} - x ^ {t} \| ^ {2} \right. \\ \left. + \frac {2 \left(p _ {\mathrm {a}} - p _ {\mathrm {a a}}\right) b ^ {2}}{n ^ {2} p _ {\mathrm {a}} ^ {2} p _ {\text {p a g e}}} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} + \left(p _ {\text {p a g e}} \left(1 - \frac {b}{p _ {\text {p a g e}}}\right) ^ {2} + (1 - p _ {\text {p a g e}})\right) \left\| h ^ {t} - \nabla f \left(x ^ {t}\right) \right\| ^ {2}\right) \\ + \rho \mathrm {E} \left(\left(\frac {2 (1 - p _ {\mathrm {a}}) \widehat {L} ^ {2}}{p _ {\mathrm {a}}} + \frac {(1 - p _ {\text {p a g e}}) L _ {\max } ^ {2}}{p _ {\mathrm {a}} B}\right) \| x ^ {t + 1} - x ^ {t} \| ^ {2} \right. \\ \left. + \left(\frac {2 \left(1 - p _ {\mathrm {a}}\right) b ^ {2}}{p _ {\mathrm {a}} p _ {\mathrm {p a g e}}} + p _ {\mathrm {p a g e}} \left(1 - \frac {b}{p _ {\mathrm {p a g e}}}\right) ^ {2} + (1 - p _ {\mathrm {p a g e}})\right) \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2}\right). \\ \end{array}
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