$$
\begin{array}{l} \mathcal {T} _ {\mathrm {i n}} \leq 4 \mathbb {E} \left[ \left\| \left(\frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\xi}} (\nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t - 1})) - \mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t - 1}) ]\right) ^ {\top} \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) \right\| _ {2} ^ {2} \right] \\ + 4 \mathbb {E} \left[ \left\| (\mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\pmb {x} ^ {t}) - \nabla g _ {\tilde {\eta}} (\pmb {x} ^ {t - 1}) ]) ^ {\top} (\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] \\ + 4 \mathbb {E} \left[ \left\| \left(\frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\xi}} \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t - 1})\right) ^ {\top} \left(\nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) - \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t - 1})\right)\right) \right. \right. \\ \left.\left. - \mathbb {E} _ {\eta | \xi} \left[ \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) - \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t - 1})\right)\right]\right)\right\| _ {2} ^ {2} \left. \right] \\ + 4 \mathbb {E} \left[ \left\| \left(\frac {1}{m} \sum_ {\tilde {\eta} \in \tilde {H} _ {\xi}} \nabla g _ {\tilde {\eta}} (\pmb {x} ^ {t - 1}) - \mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\pmb {x} ^ {t - 1}) ]\right) ^ {\top} (\nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t})\right) - \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t - 1}))\right) \right\| _ {2} ^ {2} \right] \\ \leq \frac {4 C _ {f} ^ {2}}{m} \mathbb {E} \left[ \left\| \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t - 1}) - \mathbb {E} _ {\tilde {\eta} | \xi} [ \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t}) - \nabla g _ {\tilde {\eta}} (\boldsymbol {x} ^ {t - 1}) ] \right\| _ {2} ^ {2} \right] + \frac {4 L _ {g} ^ {2} C _ {f} ^ {2}}{m} \left\| \boldsymbol {x} ^ {t} - \boldsymbol {x} ^ {t - 1} \right\| _ {2} ^ {2} \\ + 4 C _ {g} ^ {2} \mathbb {E} \left[ \left\| \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t})\right) - \nabla f _ {\xi} \left(\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\boldsymbol {x} ^ {t - 1})\right) \right. \right. \\ \left. \left. - \mathbb {E} _ {\eta | \xi} \big [ \nabla f _ {\xi} \big (\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t}) \big) - \nabla f _ {\xi} \big (\frac {1}{m} \sum_ {\eta \in H _ {\xi}} g _ {\eta} (\pmb {x} ^ {t - 1}) \big) \big ] \right\| _ {2} ^ {2} \right] \\ + \frac {4 C _ {g} ^ {4} L _ {f} ^ {2}}{m} \left\| \boldsymbol {x} ^ {t} - \boldsymbol {x} ^ {t - 1} \right\| _ {2} ^ {2}. \\ \end{array}
$$