$$
\begin{array}{l} y _ {i} f \left(\mathbf {W} ^ {\left(\widetilde {t} + 1\right)}, \mathbf {x} _ {i}\right) - y _ {i} f \left(\mathbf {W} ^ {\left(\widetilde {t}\right)}, \mathbf {x} _ {i}\right) \\ = \sum_ {j} y _ {i} j \left(F _ {j} \left(\mathbf {W} _ {j} ^ {(\tilde {t} + 1)}, \mathbf {x} _ {i}\right) - F _ {j} \left(\mathbf {W} _ {j} ^ {(\tilde {t})}, \mathbf {x} _ {i}\right)\right) \\ = \sum_ {j} y _ {i} j \cdot \frac {1}{m} \sum_ {r = 1} ^ {m} \left(\sigma \left(\left\langle \mathbf {w} _ {j, r} ^ {(\widetilde {t} + 1)}, \mathbf {x} _ {i} \right\rangle\right) - \sigma \left(\left\langle \mathbf {w} _ {j, r} ^ {(\widetilde {t})}, \mathbf {x} _ {i} \right\rangle\right)\right) \\ = \sum_ {j} y _ {i} j \cdot \frac {1}{m} \sum_ {r = 1} ^ {m} \frac {\sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle) - \sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle)}{\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle - \langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle} \cdot \left\langle - \eta \cdot \nabla_ {\mathbf {w} _ {j, r}} L _ {S} (\mathbf {W} ^ {(\widetilde {t})}), \mathbf {x} _ {i} \right\rangle \\ = \sum_ {j} y _ {i} j \cdot \frac {1}{m} \sum_ {r = 1} ^ {m} \frac {\sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle) - \sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle)}{\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle - \langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle} \cdot \left\langle \frac {\eta}{n m} \sum_ {i ^ {\prime} = 1} ^ {n} | \ell_ {i ^ {\prime}} ^ {\prime (\widetilde {t})} | \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {j, r} ^ {(\widetilde {t})}, \mathbf {x} _ {i ^ {\prime}} \rangle) \cdot j y _ {i ^ {\prime}} \mathbf {x} _ {i ^ {\prime}}, \mathbf {x} _ {i} \right\rangle \\ = \sum_ {j} \frac {1}{m} \sum_ {r = 1} ^ {m} \frac {\sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle) - \sigma (\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle)}{\langle \mathbf {w} _ {j , r} ^ {(\widetilde {t} + 1)} , \mathbf {x} _ {i} \rangle - \langle \mathbf {w} _ {j , r} ^ {(\widetilde {t})} , \mathbf {x} _ {i} \rangle} \cdot \frac {\eta}{n m} | \ell_ {i} ^ {\prime (\widetilde {t})} | \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {j, r} ^ {(\widetilde {t})}, \mathbf {x} _ {i} \rangle) \cdot \| \mathbf {x} _ {i} \| _ {2} ^ {2} \\ + \sum_ {j} \frac {1}{m} \sum_ {r = 1} ^ {m} \frac {\sigma (\langle \mathbf {w} _ {j , r} ^ {(\tilde {t} + 1)} , \mathbf {x} _ {i} \rangle) - \sigma (\langle \mathbf {w} _ {j , r} ^ {(\tilde {t})} , \mathbf {x} _ {i} \rangle)}{\langle \mathbf {w} _ {j , r} ^ {(\tilde {t} + 1)} , \mathbf {x} _ {i} \rangle - \langle \mathbf {w} _ {j , r} ^ {(\tilde {t})} , \mathbf {x} _ {i} \rangle} \cdot \frac {\eta}{n m} \sum_ {i ^ {\prime} \neq i} | \ell_ {i ^ {\prime}} ^ {\prime} (\tilde {t}) | \cdot \sigma^ {\prime} (\langle \mathbf {w} _ {j, r} ^ {(\tilde {t})}, \mathbf {x} _ {i ^ {\prime}} \rangle) \cdot \langle y _ {i ^ {\prime}} \mathbf {x} _ {i ^ {\prime}}, y _ {i} \mathbf {x} _ {i} \rangle . \\ \end{array}
$$