$$
\begin{array}{l} \mathbb {E} _ {t} \left[ F _ {1 / \bar {\rho}} \left(\mathbf {w} _ {t + 1}, \mathbf {s} _ {t + 1}, s _ {t + 1} ^ {\prime}\right) \right] \\ = F _ {1 / \bar {\rho}} (\mathbf {w} _ {t}, \mathbf {s} _ {t}, s _ {t} ^ {\prime}) + \bar {\rho} \eta \left[ \langle \hat {\mathbf {w}} _ {t} - \mathbf {w} _ {t}, \mathbb {E} _ {t} [ G _ {t} ] \rangle + \langle \hat {\mathbf {s}} _ {t} - \mathbf {s} _ {t}, \mathbb {E} _ {t} [ G _ {t} ^ {1} ] \rangle + \langle \hat {s} _ {t} ^ {\prime} - s _ {t} ^ {\prime}, \mathbb {E} _ {t} [ G _ {t} ^ {2} ] \rangle \right] \\ + \frac {3 \eta^ {2} \bar {\rho} M ^ {2}}{2} \\ \stackrel {(a)} {\leq} F _ {1 / \bar {\rho}} (\mathbf {w} _ {t}, \mathbf {s} _ {t}, s _ {t} ^ {\prime}) + \frac {3 \eta^ {2} \bar {\rho} M ^ {2}}{2} + \bar {\rho} \eta \left\{\frac {1}{n _ {+}} \sum_ {X _ {i} \in S _ {+}} \left[ F _ {i} (\hat {s} _ {t} ^ {\prime}, \hat {\mathbf {w}} _ {t}, \hat {s} _ {i, t}) - F _ {i} (s _ {t} ^ {\prime}, \mathbf {w} _ {t}, s _ {i, t}) \right. \right. \\ + C _ {f} C _ {g} \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} \left[ \| h _ {i} (\mathbf {w} _ {t}) - v _ {i, t} \| + \| h _ {j} (\mathbf {w} _ {t})) - v _ {j, t} \| \right] \\ + C _ {f} \left\| \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} g _ {i} \left(v _ {j, t} - v _ {i, t}, s _ {i, t}\right) - u _ {i, t} \right\| + C _ {f} \left\| \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} g _ {i} \left(v _ {j, t} - v _ {i, t}, s _ {i, t}\right) - u _ {i, t} \right\| \\ + \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} \rho_ {g} C _ {f} \left[ \| \left(h _ {i} (\hat {\mathbf {w}} _ {t}) - v _ {i, t} \right\| ^ {2} + \| h _ {j} (\hat {\mathbf {w}} _ {t})\right) - v _ {j, t} \| ^ {2} \bigg ] + \frac {\rho_ {g} C _ {f}}{2} \| \hat {s} _ {i, t} - s _ {i, t} \| ^ {2} \bigg ] \\ + C _ {f} C _ {g} \left[ \frac {1}{n _ {+}} \sum_ {X _ {i} \in S _ {+}} \| h _ {i} (\mathbf {w} _ {t}) - v _ {i, t} \| + \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} \| h _ {j} (\mathbf {w} _ {t}) - v _ {j, t} \| \right] + \frac {C _ {f} C _ {g} L _ {h}}{2} \| \hat {\mathbf {w}} _ {t} - \mathbf {w} _ {t} \| ^ {2} \Biggr \} \\ = F _ {1 / \bar {\rho}} (\mathbf {w} _ {t}, \mathbf {s} _ {t}, s _ {t} ^ {\prime}) + \frac {3 \eta^ {2} \bar {\rho} M ^ {2}}{2} + \bar {\rho} \eta (F (\hat {\mathbf {w}} _ {t}, \hat {\mathbf {s}} _ {t}, \hat {s} _ {t} ^ {\prime}) - F (\mathbf {w} _ {t}, \mathbf {s} _ {t}, s _ {t} ^ {\prime})) \\ + \bar {\rho} \eta \left\{\frac {1}{n _ {+}} \sum_ {X _ {i} \in S _ {+}} \left[ \frac {2 C _ {f} C _ {g}}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} \left[ \| h _ {i} (\mathbf {w} _ {t}) - v _ {i, t} \| + \| h _ {j} (\mathbf {w} _ {t})\right) - v _ {j, t} \| \right] \right. \\ + \frac {2 \rho_ {g} C _ {f}}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} \left[ \| h _ {i} (\mathbf {w} _ {t}) - v _ {i, t} \| ^ {2} + \| h _ {j} (\mathbf {w} _ {t})) - v _ {j, t} \| ^ {2} \right] + 4 \rho_ {g} C _ {f} C _ {h} \| \hat {\mathbf {w}} _ {t} - \mathbf {w} _ {t} \| ^ {2} \\ \left. + 2 C _ {f} \left\| \frac {1}{n _ {-}} \sum_ {X _ {j} \in S _ {-}} g _ {i} \left(v _ {j, t} - v _ {i, t}, s _ {i, t}\right) - u _ {i, t} \right\| + \frac {\rho_ {g} C _ {f}}{2} \| \hat {s} _ {i, t} - s _ {i, t} \| ^ {2} \right] + \frac {C _ {f} C _ {g} L _ {h}}{2} \| \hat {\mathbf {w}} _ {t} - \mathbf {w} _ {t} \| ^ {2} \Bigg \} \tag {41} \\ \end{array}
$$