$$
\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 (353) \\ = \frac {1}{4} \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} \left(0, \sigma_ {B} I _ {d \times d}\right)} \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] (354) \\ + \sum_ {l \in [ 4 ], l \neq j} \frac {1}{4} 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] (355) \\ \geq \frac {1}{4} \mu_ {j} ^ {\top} D _ {j} \left(1 - O \left(\frac {1}{(\log d) ^ {\frac {\tau}{5 0}}}\right)\right) - \sum_ {l \in [ 4 ], l \neq j} \frac {1}{4} | \mu_ {l} ^ {\top} D _ {j} | O \left(\frac {1}{d ^ {\frac {\tau}{2}}}\right) (356) \\ - \frac {1}{4} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {B} 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| (357) \\ - \sum_ {l \in [ 4 ], l \neq j} \frac {1}{4} \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| (358) \\ \geq \frac {1}{4} \sqrt {d} \left(1 - O \left(\frac {1}{(\log d) ^ {\frac {\tau}{5 0}}}\right)\right) - \frac {1}{4} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {B} I)} \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| \\ - \sum_ {l \in [ 4 ], l \neq j} \frac {1}{4} \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| (359) \\ = \frac {1}{4} \sqrt {d} \left(1 - O \left(\frac {1}{(\log d) ^ {\frac {\tau}{5 0}}}\right)\right) - \frac {1}{4} \left| \mathbb {E} _ {\mathbf {x} \sim \mathcal {N} (0, \sigma_ {B} 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| \\ - \sum_ {l \in [ 4 ], l \neq j} \frac {1}{4} \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| (360) \\ \geq \frac {1}{4} \sqrt {d} \left(1 - O \left(\frac {1}{\left(\log d\right) ^ {\frac {\tau}{5 0}}}\right)\right) - \sigma_ {B} O \left(\frac {1}{\left(\log d\right) ^ {0 . 0 1 8 \tau}}\right). (361) \\ \end{array}
$$