$$
\begin{array}{l} \left. \right. \operatorname {E} _ {k} \left[ \right. \operatorname {E} _ {p _ {\mathrm {a}}} \left[ \right. \operatorname {E} _ {p _ {\mathrm {m e g a}}} \left[ \right.\left| \right.\left| h _ {i} ^ {t + 1} - \nabla f _ {i} \left(x ^ {t + 1}\right)\right| ^ {2} \left. \right]\left. \right]\left. \right]\left. \right] \\ \left. + \left(p _ {\text {m e g a}} \left(1 - \frac {b}{p _ {\text {m e g a}}}\right) ^ {2} + \left(1 - p _ {\text {m e g a}}\right)\right) \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \right. \\ = p _ {\mathrm {m e g a}} p _ {\mathrm {a}} \mathrm {E} _ {k} \left[ \left\| h _ {i} ^ {t} + \frac {1}{p _ {\mathrm {a}}} k _ {i, 1} ^ {t + 1} - \left(h _ {i} ^ {t} + \mathrm {E} _ {k} \left[ k _ {i, 1} ^ {t + 1} \right]\right) \right\| ^ {2} \right] \\ \left. + p _ {\text {m e g a}} \left(1 - p _ {\mathrm {a}}\right) \left\| h _ {i} ^ {t} - \left(h _ {i} ^ {t} + \mathrm {E} _ {k} \left[ k _ {i, 1} ^ {t + 1} \right]\right) \right\| ^ {2} \right. \\ + \left(1 - p _ {\mathrm {m e g a}}\right) p _ {\mathrm {a}} \mathrm {E} _ {k} \left[ \left\| h _ {i} ^ {t} + \frac {1}{p _ {\mathrm {a}}} k _ {i, 2} ^ {t + 1} - \left(h _ {i} ^ {t} + \mathrm {E} _ {k} \left[ k _ {i, 2} ^ {t + 1} \right]\right) \right\| ^ {2} \right] \\ \left. + \left(1 - p _ {\text {m e g a}}\right) \left(1 - p _ {\mathrm {a}}\right) \left\| h _ {i} ^ {t} - \left(h _ {i} ^ {t} + \mathrm {E} _ {k} \left[ k _ {i, 2} ^ {t + 1} \right]\right) \right\| ^ {2} \right. \\ \left. + \left(p _ {\text {m e g a}} \left(1 - \frac {b}{p _ {\text {m e g a}}}\right) ^ {2} + (1 - p _ {\text {m e g a}})\right) \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \right. \\ = p _ {\mathrm {m e g a}} p _ {\mathrm {a}} \mathrm {E} _ {k} \left[ \left\| \frac {1}{p _ {\mathrm {a}}} k _ {i, 1} ^ {t + 1} - \mathrm {E} _ {k} \left[ k _ {i, 1} ^ {t + 1} \right] \right\| ^ {2} \right] \\ + p _ {\text {m e g a}} \left(1 - p _ {\mathrm {a}}\right) \left\| \nabla f _ {i} \left(x ^ {t + 1}\right) - \nabla f _ {i} \left(x ^ {t}\right) - \frac {b}{p _ {\text {m e g a}}} \left(h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right)\right) \right\| ^ {2} \\ + (1 - p _ {\mathrm {m e g a}}) p _ {\mathrm {a}} \mathrm {E} _ {k} \left[ \left\| \frac {1}{p _ {\mathrm {a}}} k _ {i, 2} ^ {t + 1} - \mathrm {E} _ {k} \left[ k _ {i, 2} ^ {t + 1} \right] \right\| ^ {2} \right] \\ + (1 - p _ {\mathrm {m e g a}}) (1 - p _ {\mathrm {a}}) \left\| \nabla f _ {i} \left(x ^ {t + 1}\right) - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \\ \left. + \left(p _ {\text {m e g a}} \left(1 - \frac {b}{p _ {\text {m e g a}}}\right) ^ {2} + (1 - p _ {\text {m e g a}})\right) \left\| h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \right. \\ \stackrel {(1 6)} {=} \frac {p _ {\mathrm {m e g a}}}{p _ {\mathrm {a}}} \mathrm {E} _ {k} \left[ \left| \left| k _ {i, 1} ^ {t + 1} - \mathrm {E} _ {k} \left[ k _ {i, 1} ^ {t + 1} \right] \right| \right| ^ {2} \right] \\ + \frac {\left(1 - p _ {\text {m e g a}}\right)}{p _ {\mathrm {a}}} \mathrm {E} _ {k} \left[ \left| \left| k _ {i, 2} ^ {t + 1} - \mathrm {E} _ {k} \left[ k _ {i, 2} ^ {t + 1} \right] \right| \right| ^ {2} \right] \\ + \frac {p _ {\text {m e g a}} (1 - p _ {\mathrm {a}})}{p _ {\mathrm {a}}} \left\| \nabla f _ {i} \left(x ^ {t + 1}\right) - \nabla f _ {i} \left(x ^ {t}\right) - \frac {b}{p _ {\text {m e g a}}} \left(h _ {i} ^ {t} - \nabla f _ {i} \left(x ^ {t}\right)\right) \right\| ^ {2} \\ + \frac {\left(1 - p _ {\mathrm {m e g a}}\right) \left(1 - p _ {\mathrm {a}}\right)}{p _ {\mathrm {a}}} \left\| \nabla f _ {i} \left(x ^ {t + 1}\right) - \nabla f _ {i} \left(x ^ {t}\right) \right\| ^ {2} \\ \end{array}
$$