$$
\begin{array}{l} \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \left\| \sum_ {k = 0} ^ {K - 1} \mathbf {I} _ {k} \cdot \hat {\sigma} _ {L K - 1} ^ {t ^ {\prime}} \cdot u _ {w _ {\mathcal {G} (k)}} ^ {t ^ {\prime}} \right\| ^ {2} \\ = \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \sum_ {k = 0} ^ {K - 1} \left\| \hat {\partial} _ {L _ {K - 1}} ^ {t ^ {\prime}} \cdot u _ {\mathcal {G} (k)} ^ {t ^ {\prime}} \right\| ^ {2} = \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \sum_ {k = 0} ^ {K - 1} \left\| \left(\sum_ {j = 0} ^ {K - 1} \left\langle \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right), u _ {w _ {\mathcal {G} (j)}} ^ {t ^ {\prime}} \right\rangle\right) \cdot u _ {w _ {\mathcal {G} (k)}} ^ {t ^ {\prime}} \right\| ^ {2} \\ = \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \sum_ {k = 0} ^ {K - 1} \left[ \left(\sum_ {j = 0} ^ {K - 1} \left\langle \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right), u _ {w _ {\mathcal {G} (j)}} ^ {t ^ {\prime}} \right\rangle\right) ^ {2} \cdot \left\| u _ {\mathcal {G} (k)} ^ {t ^ {\prime}} \right\| ^ {2} \right] \\ = \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \sum_ {k = 0} ^ {K - 1} \left[ \left(\left\langle \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right), u _ {w _ {\mathcal {G} (k)}} ^ {t ^ {\prime}} \right\rangle^ {2} + \sum_ {j \neq k} \left\langle \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right), u _ {w _ {\mathcal {G} (j)}} ^ {t ^ {\prime}} \right\rangle^ {2} \right. \right. \\ \left. \left. + \sum_ {m \neq n} \left\langle \nabla f _ {\mathcal {G} (m), x _ {i \left(t ^ {\prime}\right)}} \left(w ^ {t ^ {\prime} - K + m + 1}\right), u _ {w _ {\mathcal {G} (m)}} ^ {t ^ {\prime}} \right\rangle \cdot \left\langle \nabla f _ {\mathcal {G} (n), x _ {i \left(t ^ {\prime}\right)}} \left(w ^ {t ^ {\prime} - K + n + 1}\right), u _ {w _ {\mathcal {G} (n)}} ^ {t ^ {\prime}} \right\rangle\right) \cdot \left\| u _ {\mathcal {G} (k)} ^ {t ^ {\prime}} \right\| ^ {2} \right] \\ = \mathbb {E} _ {u _ {w} ^ {t ^ {\prime}}} \sum_ {k = 0} ^ {K - 1} \left[ \left(\left\langle \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right), u _ {w _ {\mathcal {G} (k)}} ^ {t ^ {\prime}} \right\rangle^ {2} + \sum_ {j \neq k} \left\langle \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right), u _ {w _ {\mathcal {G} (j)}} ^ {t ^ {\prime}} \right\rangle^ {2}\right) \cdot \left\| u _ {\mathcal {G} (k)} ^ {t ^ {\prime}} \right\| ^ {2} \right] \\ \leq \sum_ {k = 0} ^ {K - 1} \left[ \right.\left( \right.d _ {k} + 4) \left\| \nabla f _ {\mathcal {G} (k), x _ {i \left(t ^ {\prime}\right)}} \left(w ^ {t ^ {\prime} - K + k + 1}\right)\right\| ^ {2} + \sum_ {j \neq k} \left(d _ {k} \left\| \nabla f _ {\mathcal {G} (j), x _ {i \left(t ^ {\prime}\right)}} \left(w ^ {t ^ {\prime} - K + j + 1}\right)\right\| ^ {2}\right)\left. \right] \\ = \sum_ {k = 0} ^ {K - 1} \left[ d _ {k} \sum_ {j = 0} ^ {K - 1} \left\| \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right) \right\| ^ {2} + 4 \left\| \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right) \right\| ^ {2} \right] \\ = \left(\sum_ {k = 0} ^ {K - 1} d _ {k}\right) \sum_ {j = 0} ^ {K - 1} \left\| \nabla f _ {\mathcal {G} (j), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + j + 1}\right) \right\| ^ {2} + 4 \sum_ {k = 0} ^ {K - 1} \left\| \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right) \right\| ^ {2} \\ = (d + 4) \sum_ {k = 0} ^ {K - 1} \left\| \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right) \right\| ^ {2} = (d + 4) \left\| \sum_ {k = 0} ^ {K - 1} \mathbf {I} _ {k} \nabla f _ {\mathcal {G} (k), x _ {i (t ^ {\prime})}} \left(w ^ {t ^ {\prime} - K + k + 1}\right) \right\| ^ {2}, \\ \end{array}
$$