$$
\begin{array}{l} = - \gamma_ {1} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \langle \mathbb {E} _ {\boldsymbol {\eta} ^ {k}} [ \bar {\boldsymbol {g}} _ {\boldsymbol {\eta} ^ {k}} (\widetilde {\boldsymbol {x}} ^ {k}) ], \boldsymbol {x} ^ {k} - \gamma_ {1} \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) - \boldsymbol {x} ^ {*} \rangle \right] \\ - \gamma_ {1} ^ {2} \mathbb {E} \left[ \langle \overline {{\boldsymbol {g}}} _ {\boldsymbol {\eta} ^ {k}} (\widetilde {\boldsymbol {x}} ^ {k}), \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) \rangle \right] \\ \stackrel {(1 4)} {=} - \gamma_ {1} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \langle F (\boldsymbol {x} ^ {k} - \gamma_ {1} \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k})), \boldsymbol {x} ^ {k} - \gamma_ {1} \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) - \boldsymbol {x} ^ {*} \rangle \right] \\ - \frac {\gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}, \boldsymbol {\eta} ^ {k}} \left[ \left\| \overline {{\boldsymbol {g}}} _ {\boldsymbol {\eta} ^ {k}} \left(\widetilde {\boldsymbol {x}} ^ {k}\right) \right\| ^ {2} \right] - \frac {\gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \left\| \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} \left(\boldsymbol {x} ^ {k}\right) \right\| ^ {2} \right] \\ + \frac {\gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}, \boldsymbol {\eta} ^ {k}} \left[ \left\| \overline {{\boldsymbol {g}}} _ {\boldsymbol {\eta} ^ {k}} \left(\widetilde {\boldsymbol {x}} ^ {k}\right) - \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} \left(\boldsymbol {x} ^ {k}\right) \right\| ^ {2} \right] \\ \stackrel {\text {(Q S M), (1 6)}} {\leq} \quad - \mu \gamma_ {1} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}, \boldsymbol {\eta} ^ {k}} \left[ \left\| \boldsymbol {x} ^ {k} - \boldsymbol {x} ^ {*} - \gamma_ {1} \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) \right\| ^ {2} \right] - \frac {\gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \left\| \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) \right\| ^ {2} \right] \\ + \frac {4 \gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \left\| \overline {{\boldsymbol {g}}} _ {\boldsymbol {\xi} ^ {k}} \left(\boldsymbol {x} ^ {k}\right) - \widehat {\boldsymbol {g}} _ {\boldsymbol {\xi} ^ {k}} \left(\boldsymbol {x} ^ {k}\right) \right\| ^ {2} \right] \\ + \frac {4 \gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \left\| F (\boldsymbol {x} ^ {k}) - F (\widetilde {\boldsymbol {x}} ^ {k}) \right\| ^ {2} \right] \\ + \frac {4 \gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}} \left[ \left\| \overline {{\boldsymbol {g}}} _ {\boldsymbol {\xi} ^ {k}} (\boldsymbol {x} ^ {k}) - F (\boldsymbol {x} ^ {k}) \right\| ^ {2} \right] \\ + \frac {4 \gamma_ {1} ^ {2}}{2} \mathbb {E} _ {\boldsymbol {\xi} ^ {k}, \boldsymbol {\eta} ^ {k}} \left[ \left\| \overline {{\boldsymbol {g}}} _ {\boldsymbol {\eta} ^ {k}} \left(\widetilde {\boldsymbol {x}} ^ {k}\right) - F \left(\widetilde {\boldsymbol {x}} ^ {k}\right) \right\| ^ {2} \right] \\ \end{array}
$$