$$
\begin{array}{l} \left\langle \mathbb {E} _ {(\mathbf {x}, y)} \left[ y \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} \right\rangle - \mathbf {b} _ {i}\right) \mathbf {x} \right], y _ {(j)} D _ {j} \right\rangle (281) \\ = p _ {j} \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {j} I _ {d \times d})} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {j} \right\rangle - \mathbf {b} _ {i}\right) (\mathbf {x} + \mu_ {j}) ^ {\top} D _ {j} \right] (282) \\ + \sum_ {l \in [ r ], l \neq j} p _ {l} y _ {(l)} y _ {(j)} \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {l} I _ {d \times d})} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {l} \right\rangle - \mathbf {b} _ {i}\right) (\mathbf {x} + \mu_ {l}) ^ {\top} D _ {j} \right] (283) \\ \geq p _ {j} \mu_ {j} ^ {\top} D _ {j} \left(1 - O \left(\frac {1}{d ^ {\tau}}\right)\right) - \sum_ {l \in [ r ], l \neq j} p _ {l} | \mu_ {l} ^ {\top} D _ {j} | O \left(\frac {1}{d ^ {\tau}}\right) (284) \\ - p _ {j} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {j} I)} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {j} \right\rangle - \mathbf {b} _ {i}\right) \mathbf {x} ^ {\top} D _ {j} \right] \right| (285) \\ - \sum_ {l \in [ r ], l \neq j} p _ {l} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {l} I)} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {l} \right\rangle - \mathbf {b} _ {i}\right) \mathbf {x} ^ {\top} D _ {j} \right] \right| (286) \\ \geq p _ {j} B _ {\mu 1} \sqrt {d} \left(1 - O \left(\frac {1}{d ^ {\tau}}\right)\right) - B _ {\mu 2} O \left(\frac {1}{d ^ {\tau - \frac {1}{2}}}\right) (287) \\ \left. \right. - p _ {j} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} \left(0, \sigma_ {j} I\right)} \left[\left(1 - \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {j} \right\rangle - \mathbf {b} _ {i}\right) - 1\right) \mathbf {x} ^ {\top} D _ {j} \right]\right| (288) \\ - \sum_ {l \in [ r ], l \neq j} p _ {l} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {l} I)} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {l} \right\rangle - \mathbf {b} _ {i}\right) \mathbf {x} ^ {\top} D _ {j} \right] \right| (289) \\ = p _ {j} B _ {\mu 1} \sqrt {d} \left(1 - O \left(\frac {1}{d ^ {\tau}}\right)\right) - B _ {\mu 2} O \left(\frac {1}{d ^ {\tau - \frac {1}{2}}}\right) (290) \\ - p _ {j} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {j} I)} \left[ \left(1 - \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {j} \right\rangle - \mathbf {b} _ {i}\right)\right) \mathbf {x} ^ {\top} D _ {j} \right] \right| (291) \\ - \sum_ {l \in [ r ], l \neq j} p _ {l} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {l} I)} \left[ \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {i} ^ {(0)}, \mathbf {x} + \mu_ {l} \right\rangle - \mathbf {b} _ {i}\right) \mathbf {x} ^ {\top} D _ {j} \right] \right| (292) \\ \geq p _ {j} B _ {\mu 1} \sqrt {d} \left(1 - O \left(\frac {1}{d ^ {\tau}}\right)\right) - B _ {\mu 2} O \left(\frac {1}{d ^ {\tau - \frac {1}{2}}}\right) - \sigma_ {B +} O \left(\frac {1}{d ^ {0 . 9 \tau}}\right). (293) \\ \end{array}
$$