$$
\begin{array}{l} \left. \left| \boldsymbol {a} ^ {\top} \boldsymbol {P} _ {1} \boldsymbol {a} \right| = \left| \mathbb {E} _ {\boldsymbol {x}} \mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {1}} ^ {\star T} \boldsymbol {h} ^ {\star}\right) \left(\mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star T} \boldsymbol {h} ^ {\star}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h} ^ {\star}\right) ^ {2} - \mathcal {J} _ {\ell , k} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\top} \boldsymbol {h}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2}\right) \right| \right. \\ \leq \mathbb {E} _ {\boldsymbol {x}} \left| \mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star T} \boldsymbol {h} ^ {\star}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h} ^ {\star}\right) ^ {2} - \mathcal {J} _ {\ell , k} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\top} \boldsymbol {h}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} \right| \\ \leq \mathbb {E} _ {\boldsymbol {x}} \left| \mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star T} \boldsymbol {h} ^ {\star}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h} ^ {\star}\right) ^ {2} - \mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h} ^ {\star}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} \right| \\ + \mathbb {E} _ {\boldsymbol {x}} \left| \mathcal {J} _ {\ell , k} ^ {\star} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h} ^ {\star}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} - \mathcal {J} _ {\ell , k} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} \right| \tag {118} \\ + \mathbb {E} _ {\boldsymbol {x}} \left| \mathcal {J} _ {\ell , k} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} - \mathcal {J} _ {\ell , k} \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\top} \boldsymbol {h}\right) \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} \right| \\ \lesssim \| \boldsymbol {W} - \boldsymbol {W} ^ {\star} \| _ {2} + \| \boldsymbol {W} - \boldsymbol {W} ^ {\star} \| _ {2} \\ + \mathbb {E} _ {\boldsymbol {x}} \left| \left(\phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h}\right) - \phi^ {\prime} \left(\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h}\right)\right) \cdot \left(\boldsymbol {a} ^ {\top} \boldsymbol {h}\right) ^ {2} \right| \\ \lesssim \| \boldsymbol {W} - \boldsymbol {W} ^ {\star} \| _ {2} + \mathbb {E} _ {\boldsymbol {x}} \left| \left(\phi^ {\prime} (\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h}) - \phi^ {\prime} (\boldsymbol {w} _ {\ell , j _ {2}} ^ {\star \top} \boldsymbol {h})\right) \cdot (\boldsymbol {a} ^ {\top} \boldsymbol {h}) ^ {2} \right|. \\ \end{array}
$$