$$
\begin{array}{l} \underbrace {\frac {1}{m} \sum_ {i = 1} ^ {m - m _ {0}} \left( \begin{array}{c} \boldsymbol {x} ^ {t} - \eta_ {\boldsymbol {x} , t} \nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , j _ {t} (i)}) - \left(\dot {\boldsymbol {x}} ^ {t} - \eta_ {\boldsymbol {x} , t} \nabla_ {\boldsymbol {x}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , j _ {t} (i)}\right) \\ \boldsymbol {y} ^ {t} + \eta_ {\boldsymbol {y}, t} \nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , j _ {t} (i)}) - \left(\dot {\boldsymbol {y}} ^ {t} + \eta_ {\boldsymbol {y}, t} \nabla_ {\boldsymbol {y}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , j _ {t} (i)})\right) \end{array} \right)} _ {I _ {1}} \\ + \underbrace {\frac {1}{m} \sum_ {i = m - m _ {0} + 1} ^ {m} \left( \begin{array}{c} \boldsymbol {x} ^ {t} - \eta_ {\boldsymbol {x} , t} \nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , n}) - \left(\dot {\boldsymbol {x}} ^ {t} - \eta_ {\boldsymbol {x} , t} \nabla_ {\boldsymbol {x}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , n} ^ {\prime})\right) \\ \boldsymbol {y} ^ {t} + \eta_ {\boldsymbol {y}, t} \nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , n}) - \left(\dot {\boldsymbol {y}} ^ {t} + \eta_ {\boldsymbol {y}, t} \nabla_ {\boldsymbol {y}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , n} ^ {\prime})\right) \end{array} \right)} _ {I _ {2}} \\ + \underbrace {\frac {1}{m} \sum_ {i = 1} ^ {m - m _ {0}} \left( \begin{array}{c} - \eta_ {\boldsymbol {x} , t} \left(\nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} _ {i} ^ {t} , \boldsymbol {y} _ {i} ^ {t} ; \xi_ {i , j _ {t} (i)}) - \nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , j _ {t} (i)})\right) \\ \eta_ {\boldsymbol {y}, t} \left(\nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} _ {i} ^ {t} , \boldsymbol {y} _ {i} ^ {t} ; \xi_ {i , j _ {t} (i)}) - \nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , j _ {t} (i)})\right) \end{array} \right)} _ {I I _ {1}} \\ - \underbrace {\frac {1}{m} \sum_ {i = 1} ^ {m - m _ {0}} \left( \begin{array}{c} - \eta_ {\boldsymbol {x} , t} \left(\nabla_ {\boldsymbol {x}} f _ {i} \left(\dot {\boldsymbol {x}} _ {t} ^ {t}, \dot {\boldsymbol {y}} _ {i} ^ {t}; \xi_ {i , j _ {t} (i)}\right) - \nabla_ {\boldsymbol {x}} f _ {i} \left(\dot {\boldsymbol {x}} ^ {t}, \dot {\boldsymbol {y}} ^ {t}; \xi_ {i , j _ {t} (i)}\right)\right) \\ \eta_ {\boldsymbol {y}, t} \left(\nabla_ {\boldsymbol {y}} f _ {i} \left(\dot {\boldsymbol {x}} _ {t} ^ {t}, \dot {\boldsymbol {y}} _ {i} ^ {t}; \xi_ {i, j _ {t} (i)}\right) - \nabla_ {\boldsymbol {y}} f _ {i} \left(\dot {\boldsymbol {x}} ^ {t}, \dot {\boldsymbol {y}} ^ {t}; \xi_ {i, j _ {t} (i)}\right)\right) \end{array} \right)} _ {I I _ {2}} \tag {7} \\ + \underbrace {\frac {1}{m} \sum_ {i = m - m _ {0} + 1} ^ {m} \left( \begin{array}{c} - \eta_ {\boldsymbol {x} , t} \left(\nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} _ {i} ^ {t} , \boldsymbol {y} _ {i} ^ {t} ; \xi_ {i , n}) - \nabla_ {\boldsymbol {x}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , n})\right) \\ \eta_ {\boldsymbol {y}, t} \left(\nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} _ {i} ^ {t} , \boldsymbol {y} _ {i} ^ {t} ; \xi_ {i , n}) - \nabla_ {\boldsymbol {y}} f _ {i} (\boldsymbol {x} ^ {t} , \boldsymbol {y} ^ {t} ; \xi_ {i , n})\right) \end{array} \right)} _ {I I _ {1} ^ {\prime}} \\ - \underbrace {\frac {1}{m} \sum_ {i = m - m _ {0} + 1} ^ {m} \left( \begin{array}{c} - \eta_ {\boldsymbol {x} , t} \left(\nabla_ {\boldsymbol {x}} f _ {i} (\dot {\boldsymbol {x}} _ {i} ^ {t} , \dot {\boldsymbol {y}} _ {i} ^ {t} ; \xi_ {i , n} ^ {\prime}) - \nabla_ {\boldsymbol {x}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , n} ^ {\prime})\right) \\ \eta_ {\boldsymbol {y}, t} \left(\nabla_ {\boldsymbol {y}} f _ {i} (\dot {\boldsymbol {x}} _ {i} ^ {t} , \dot {\boldsymbol {y}} _ {i} ^ {t} ; \xi_ {i , n} ^ {\prime}) - \nabla_ {\boldsymbol {y}} f _ {i} (\dot {\boldsymbol {x}} ^ {t} , \dot {\boldsymbol {y}} ^ {t} ; \xi_ {i , n} ^ {\prime})\right) \end{array} \right)} _ {I I _ {2} ^ {\prime}} \\ \end{array}
$$