$$
\begin{array}{l} \mathbb {E} _ {\boldsymbol {W}, \boldsymbol {U}} \left[ Z _ {t, k} ^ {2} (\boldsymbol {W}, \boldsymbol {U}) \right] \\ = \mathbb {E} _ {\boldsymbol {W}, \boldsymbol {U}} \left[ \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \left| s _ {t, \boldsymbol {\theta}, k} (\boldsymbol {x}) - \bar {s} _ {t, \bar {\boldsymbol {\theta}}, k} (\boldsymbol {x}) \right| ^ {2} \right] \right] \\ \stackrel {(v)} {=} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {\boldsymbol {W}, \boldsymbol {U}} \left[ \left| s _ {t, \boldsymbol {\theta}, k} (\boldsymbol {x}) - \bar {s} _ {t, \bar {\boldsymbol {\theta}}, k} (\boldsymbol {x}) \right| ^ {2} \right] \right] \\ = \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {\boldsymbol {W}, \boldsymbol {U}} \left[ \left| \sum_ {i = 1} ^ {m} \left(a _ {i, k} \sigma (\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {i} ^ {\top} \boldsymbol {e} (t)) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma (\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)) \right]\right) \right| ^ {2} \right] \right] \\ = \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {\boldsymbol {W}, U} \left[ \sum_ {i = 1} ^ {m} \left(a _ {i, k} \sigma \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {i} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {\left(\boldsymbol {w}, \boldsymbol {u}\right) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) \right]\right) ^ {2} \right] \right] \\ + \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \right. \mathbb {E} _ {\boldsymbol {W}, \boldsymbol {U}} \left[ \sum_ {i \neq j} \left(a _ {i, k} \sigma \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {i} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right)\right]\right.\right. \\ \left. \times \left(a _ {j, k} \sigma \left(\boldsymbol {w} _ {j} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {j} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) \right]\right) \right] \\ = \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \sum_ {i = 1} ^ {m} \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) \right]\right) ^ {2} \right] \right] \\ + \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \sum_ {i \neq j} \mathbb {E} _ {(\boldsymbol {w} _ {i}, \boldsymbol {u} _ {i}) \sim \rho_ {0}} \left[ \left(a _ {i, k} \sigma (\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {i} ^ {\top} \boldsymbol {e} (t)) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} [ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma (\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)) ]\right) \right] \right. \\ \left. \times \mathbb {E} _ {\left(\boldsymbol {w} _ {j}, \boldsymbol {u} _ {j}\right) \sim \rho_ {0}} \left[ \left(a _ {j, k} \sigma \left(\boldsymbol {w} _ {j} ^ {\top} \boldsymbol {x} + \boldsymbol {u} _ {j} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {\left(\boldsymbol {w}, \boldsymbol {u}\right) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) \right]\right) \right] \right] \\ = \frac {1}{m ^ {2}} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \sum_ {i = 1} ^ {m} \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) - \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right) \right]\right) ^ {2} \right] \right] \\ \leq \frac {1}{m} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(a _ {k} (\boldsymbol {w}, \boldsymbol {u}) \sigma \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} + \boldsymbol {u} ^ {\top} \boldsymbol {e} (t)\right)\right) ^ {2} \right] \right] \\ \leq \frac {1}{m} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(| a _ {k} (\boldsymbol {w}, \boldsymbol {u}) | \left(\| \boldsymbol {w} \| _ {1} C _ {T, \delta} + \| \boldsymbol {u} \| _ {1} \| \boldsymbol {e} (t) \| _ {\infty}\right)\right) ^ {2} \right] \right] \\ \leq \frac {1}{m} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(| a _ {k} (\boldsymbol {w}, \boldsymbol {u}) | (C _ {T, \delta} + \| \boldsymbol {e} (t) \| _ {\infty})\right) ^ {2} \right] \right] \\ \stackrel {\text {(i v)}} {\lesssim} \frac {1}{m} \mathbb {E} _ {\boldsymbol {x} \sim p _ {t}} \left[ \mathbb {E} _ {(\boldsymbol {w}, \boldsymbol {u}) \sim \rho_ {0}} \left[ \left(C _ {T, \delta} + C _ {T, e}\right) ^ {2} \right] \right] (31) \\ \lesssim \left(C _ {T, \delta} + C _ {T, e}\right) ^ {2} \frac {1}{m}, (32) \\ \end{array}
$$