$$
\begin{array}{l} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \bar {\boldsymbol {x}} _ {r + 1} - \bar {\boldsymbol {x}} _ {r} \right\| ^ {2} \\ = \mathbb {1} _ {\mathcal {A} _ {r}} \eta^ {2} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \boldsymbol {g} _ {r, k} ^ {i} \right\| ^ {2} \\ = \mathbb {1} _ {\mathcal {A} _ {r}} \eta^ {2} \left\| \frac {1}{S} \sum_ {i \in S _ {r}} \sum_ {k = 0} ^ {I - 1} \nabla F _ {i} \left(\boldsymbol {x} _ {r, k} ^ {i}; \xi_ {r, k} ^ {i}\right) - \boldsymbol {G} _ {r} ^ {i} + \boldsymbol {G} _ {r} \right\| ^ {2} \\ = \mathbb {1} _ {\mathcal {A} _ {r}} \eta^ {2} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \left(\nabla f _ {i} \left(\boldsymbol {x} _ {r, k} ^ {i}\right) - \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r})\right) + \left(\nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) - \boldsymbol {G} _ {r} ^ {i}\right) + \left(\boldsymbol {G} _ {r} - \nabla f (\bar {\boldsymbol {x}} _ {r})\right) + \nabla f (\bar {\boldsymbol {x}} _ {r}) \right\| ^ {2} \\ \leq 4 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \| \nabla f (\bar {\boldsymbol {x}} _ {r}) \| ^ {2} + 4 \eta^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \nabla F _ {i} \left(\boldsymbol {x} _ {r, k} ^ {i}; \xi_ {r, k} ^ {i}\right) - \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) \right\| ^ {2} \\ + 4 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) - \boldsymbol {G} _ {r} ^ {i} \right\| ^ {2} + 4 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \| \boldsymbol {G} _ {r} - \nabla f (\bar {\boldsymbol {x}} _ {r}) \| ^ {2} \\ \leq 4 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \| \nabla f (\bar {\boldsymbol {x}} _ {r}) \| ^ {2} + 8 \eta^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \nabla f _ {i} (\boldsymbol {x} _ {r, k} ^ {i}) - \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) \right\| ^ {2} \\ + 8 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S I} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) - \nabla f _ {i} (\boldsymbol {y} _ {r, k} ^ {i}) \right\| ^ {2} \\ + 8 \eta^ {2} I ^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{N I} \sum_ {i = 1} ^ {N} \sum_ {k = 0} ^ {I - 1} \nabla f _ {i} \left(\boldsymbol {y} _ {r, k} ^ {i}\right) - \nabla f _ {i} (\bar {\boldsymbol {x}} _ {r}) \right\| ^ {2} \\ + 8 \eta^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \boldsymbol {\epsilon} _ {r, k} ^ {i} \right\| ^ {2} + 8 \eta^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{S} \sum_ {i \in \mathcal {S} _ {r}} \sum_ {k = 0} ^ {I - 1} \tilde {\boldsymbol {\epsilon}} _ {r, k} ^ {i} \right\| ^ {2} + 8 \eta^ {2} \mathbb {1} _ {\mathcal {A} _ {r}} \left\| \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {k = 0} ^ {I - 1} \tilde {\boldsymbol {\epsilon}} _ {r, k} ^ {i} \right\| ^ {2}, \tag {38} \\ \end{array}
$$