$$
\begin{array}{l} \langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t}, 0)}, \boldsymbol {\xi} _ {i} \rangle = \langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t} - 1, 0)}, \boldsymbol {\xi} _ {i} \rangle - \frac {\eta}{B m} \cdot \sum_ {b ^ {\prime} = 0} ^ {H - 1} \sum_ {i ^ {\prime} \in \mathcal {I} _ {\widetilde {t} - 1, b ^ {\prime}}} \ell_ {i ^ {\prime}} ^ {(\widetilde {t} - 1, \widetilde {b})} \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t} - 1, b ^ {\prime})}, y _ {i ^ {\prime}} \boldsymbol {\mu} \rangle) \cdot \langle y _ {i ^ {\prime}} \boldsymbol {\mu}, \boldsymbol {\xi} _ {i} \rangle y _ {i ^ {\prime}} \\ - \frac {\eta (P - 1)}{B m} \cdot \sum_ {b ^ {\prime} = 0} ^ {H - 1} \sum_ {i ^ {\prime} \in \mathcal {I} _ {\widetilde {t} - 1, b ^ {\prime}}} \ell_ {i ^ {\prime}} ^ {(\widetilde {t} - 1, b ^ {\prime})} \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t} - 1, \widetilde {b})}, \boldsymbol {\xi} _ {i ^ {\prime}} \rangle) \cdot \langle \boldsymbol {\xi} _ {i ^ {\prime}}, \boldsymbol {\xi} _ {i} \rangle \\ = \left\langle \mathbf {w} _ {y _ {i}, r} ^ {\left(\widetilde {t} - 1, 0\right)}, \boldsymbol {\xi} _ {i} \right\rangle - \frac {\eta}{B m} \cdot \sum_ {b ^ {\prime} = 0} ^ {H - 1} \sum_ {i ^ {\prime} \in \mathcal {I} _ {\widetilde {t} - 1, b ^ {\prime}}} \ell_ {i ^ {\prime}} ^ {\left(\widetilde {t} - 1, b ^ {\prime}\right)} \cdot \sigma^ {\prime} \left(\left\langle \mathbf {w} _ {y _ {i}, r} ^ {\left(\widetilde {t} - 1, b ^ {\prime}\right)}, \widehat {y} _ {i ^ {\prime}} \boldsymbol {\mu} \right\rangle\right) \cdot \left\langle y _ {i ^ {\prime}} \boldsymbol {\mu}, \boldsymbol {\xi} _ {i} \right\rangle y _ {i ^ {\prime}} \\ - \frac {\eta (P - 1)}{B m} \cdot \ell_ {i} ^ {(\widetilde {t} - 1, b _ {i} ^ {(\widetilde {t} - 1)})} \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t} - 1, \widetilde {b})}, \boldsymbol {\xi} _ {i} \rangle) \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {2} \\ - \frac {\eta (P - 1)}{B m} \cdot \sum_ {b ^ {\prime} = 0} ^ {H - 1} \sum_ {i ^ {\prime} \in \mathcal {I} _ {\tilde {t} - 1, b ^ {\prime}}} \ell_ {i ^ {\prime}} ^ {(\tilde {t}, b ^ {\prime})} \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {y _ {i}, r} ^ {(\tilde {t}, b ^ {\prime})}, \boldsymbol {\xi} _ {i ^ {\prime}} \rangle) \cdot \langle \boldsymbol {\xi} _ {i ^ {\prime}}, \boldsymbol {\xi} _ {i} \rangle   \mathbb {1} (i ^ {\prime} \neq i) \\ = \left\langle \mathbf {w} _ {y _ {i}, r} ^ {(\widetilde {t}, 0)}, \boldsymbol {\xi} _ {i} \right\rangle - \frac {\eta (P - 1)}{B m} \cdot \underbrace {\ell_ {i} ^ {(\widetilde {t} - 1 , b _ {i} ^ {(\widetilde {t} - 1)})} \cdot \| \boldsymbol {\xi} _ {i} \| _ {2} ^ {2}} _ {\mathrm {I} _ {4}} \\ - \frac {\eta (P - 1)}{B m} \cdot \underbrace {\sum_ {b ^ {\prime} = 0} ^ {H - 1} \sum_ {i ^ {\prime} \in \mathcal {I} _ {\widetilde {t} - 1 , b ^ {\prime}}} \ell_ {i ^ {\prime}} ^ {(\widetilde {t} - 1 , b ^ {\prime})} \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {y _ {i} , r} ^ {(\widetilde {t} - 1 , b ^ {\prime})}, \boldsymbol {\xi} _ {i ^ {\prime}} \rangle) \cdot \langle \boldsymbol {\xi} _ {i ^ {\prime}} , \boldsymbol {\xi} _ {i} \rangle \mathbb {1} (i ^ {\prime} \neq i)} _ {\mathrm {I} _ {5}} \\ \end{array}
$$