$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} \neq \tilde {v} _ {t} ^ {u}, p _ {1} = v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = v _ {m}, \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {\tilde {v} _ {t} ^ {u}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \geq \beta \right] \\ \leq \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {h = 1} ^ {H} w _ {h} \left[ \tilde {\eta} _ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \epsilon \right] \leq \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta} \right] \\ = \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \tilde {\eta} _ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta} + \epsilon \right] \\ = \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {v _ {j} \in \nu} \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta} + \epsilon \right] \\ \leq \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {m} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \frac {\frac {\sum_ {h = 1} ^ {H} [ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) ]}{\beta} - \sum_ {v _ {j} \neq v _ {m}} \left[ \tau_ {v _ {j} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]}{\tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} + \frac {\epsilon}{\tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} \right] \\ \leq \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \eta_ {m} (\tilde {\mathbf {x}} _ {t} ^ {u}) \leq \frac {\frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]}{\beta_ {2} - \xi_ {2}} - \sum_ {v _ {j} \neq v _ {m}} \left[ \tau_ {v _ {j} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]}{\tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} + \frac {\epsilon}{\tau_ {v _ {m} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u})} \right] \\ = \mathbb {P} \left[ p _ {1} = v _ {m}, \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \frac {\frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta_ {2}} - \sum_ {v _ {j} \neq v _ {m}} \left[ \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\tau_ {v _ {m} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} + \frac {\epsilon}{\tau_ {v _ {m} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right. \\ \left. + \frac {\xi_ {2} \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\beta_ {2} \left(\beta_ {2} - \xi_ {2}\right) \tau_ {v _ {m} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right)} \right] \tag {11} \\ \end{array}
$$