$$
\begin{array}{l} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi , \tilde {\eta} | \xi} [ \| G ^ {t + 1} - \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi , \tilde {\eta} | \xi} [ G ^ {t + 1} ] \right\| _ {2} ^ {2} ] \right] \\ = \mathbb {E} _ {\boldsymbol {\xi}} \left[ \mathbb {E} _ {\eta | \boldsymbol {\xi}, \tilde {\eta} | \boldsymbol {\xi}} \left[ \left\| \left(\frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\boldsymbol {\xi}}} \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \mathbb {E} _ {\tilde {\eta} | \boldsymbol {\xi}} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) ]\right) ^ {\top} \nabla f _ {\boldsymbol {\xi}} \left(\frac {1}{m} \sum_ {\eta \in H _ {\boldsymbol {\xi}}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) \right\| _ {2} ^ {2} \right] \right] \\ + \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi} \left[ \left\| \left(\mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) ]\right) ^ {\top} \left(\nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) - \mathbb {E} _ {\eta | \xi} [ \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) ]\right) \right\| _ {2} ^ {2} \right] \right] \\ \leq C _ {f} ^ {2} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\tilde {\eta} | \xi} \left[ \left\| \frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\xi}} \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) ] \right\| _ {2} ^ {2} \right] \right] \\ + C _ {g} ^ {2} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi} \left[ \left\| \nabla f _ {\xi} (\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t})) - \mathbb {E} _ {\eta | \xi} [ \nabla f _ {\xi} (\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t})) ] \right\| _ {2} ^ {2} \right] \right] \\ = C _ {f} ^ {2} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\tilde {\eta} | \xi} \left[ \left\| \frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\xi}} \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) ] \right\| _ {2} ^ {2} \right] \right] \\ + C _ {g} ^ {2} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi} \left[ \left\| \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) - \nabla f _ {\xi} \left(\mathbb {E} _ {\eta} [ g _ {\eta} (\boldsymbol {x} ^ {t}) ]\right) \right\| _ {2} ^ {2} \right] \right] \\ - C _ {g} ^ {2} \mathbb {E} _ {\xi} \left[ \left\| \nabla f _ {\xi} \left(\mathbb {E} _ {\eta} \left[ g _ {\eta} (\boldsymbol {x} ^ {t}) \right]\right) - \mathbb {E} _ {\eta | \xi} \left[ \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) \right] \right\| _ {2} ^ {2} \right] \\ \leq \frac {\zeta_ {g} ^ {2} C _ {f} ^ {2}}{m} + C _ {g} ^ {2} L _ {f} \mathbb {E} _ {\xi} \left[ \mathbb {E} _ {\eta | \xi} \left[ \left\| \frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t}) - \mathbb {E} _ {\eta | \xi} [ g _ {\eta} (\pmb {x} ^ {t}) ] \right\| _ {2} ^ {2} \right] \right] \\ \leq \frac {C _ {f} ^ {2} \zeta_ {g} ^ {2}}{m} + \frac {C _ {g} ^ {2} L _ {f}}{m} \mathbb {E} _ {\boldsymbol {\xi}} \left[ \mathbb {E} _ {\boldsymbol {\eta} | \boldsymbol {\xi}} \left[ \left\| g _ {\boldsymbol {\eta}} (\boldsymbol {x} ^ {t}) - \mathbb {E} _ {\boldsymbol {\eta} | \boldsymbol {\xi}} [ g _ {\boldsymbol {\eta}} (\boldsymbol {x} ^ {t}) ] \right\| _ {2} ^ {2} \right] \right] \\ \leq \frac {\zeta_ {g} ^ {2} C _ {f} ^ {2} + \sigma_ {g} ^ {2} C _ {g} ^ {2} L _ {f} ^ {2}}{m} = \frac {\sigma_ {\mathrm {i n}} ^ {2}}{m}. \\ \end{array}
$$