$$
\begin{array}{l} \left\| \boldsymbol {U} _ {i, n} ^ {t ^ {\prime}} - \boldsymbol {U} _ {i, n} ^ {t} \right\| _ {2} \\ \leq \left\| \boldsymbol {W} _ {K} ^ {t ^ {\prime}} \boldsymbol {X} _ {n} ^ {\top} \left(\operatorname {d i a g} \left(\boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}}\right) - \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} ^ {\top}\right) - \boldsymbol {W} _ {K} ^ {t} \boldsymbol {X} _ {n} ^ {\top} \left(\operatorname {d i a g} \left(\boldsymbol {\beta} _ {i, n} ^ {t}\right) - \boldsymbol {\beta} _ {i, n} ^ {t} \boldsymbol {\beta} _ {i, n} ^ {t} ^ {\top}\right) \right\| _ {2} \| \boldsymbol {X} _ {n} \| _ {2} \\ \leq C _ {x} \sqrt {d _ {s}} \left[ \left\| \boldsymbol {W} _ {K} ^ {t ^ {\prime}} - \boldsymbol {W} _ {K} ^ {t} \right\| _ {2} \| \boldsymbol {X} _ {n} \| _ {2} \left\| \operatorname {d i a g} \left(\boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}}\right) - \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} ^ {\top} \right\| _ {2} \right. \\ \left. + \left\| \boldsymbol {W} _ {K} ^ {t} \right\| _ {2} \| \boldsymbol {X} _ {n} \| _ {2} \left\| \operatorname {d i a g} \left(\beta_ {i, n} ^ {t ^ {\prime}}\right) - \beta_ {i, n} ^ {t ^ {\prime}} \beta_ {i, n} ^ {t ^ {\prime}} ^ {\top} - \operatorname {d i a g} \left(\beta_ {i, n} ^ {t}\right) - \beta_ {i, n} ^ {t} \beta_ {i, n} ^ {t} ^ {\top} \right\| _ {2} \right] \\ \leq C _ {x} ^ {2} d _ {s} \left[ \left\| \boldsymbol {\theta} ^ {t ^ {\prime}} - \boldsymbol {\theta} ^ {t} \right\| _ {2} + \bar {\lambda} _ {K} \left(\left\| \operatorname {d i a g} \left(\beta_ {i, n} ^ {t ^ {\prime}}\right) - \operatorname {d i a g} \left(\beta_ {i, n} ^ {t}\right) \right\| _ {2} + \left\| \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} ^ {\top} - \boldsymbol {\beta} _ {i, n} ^ {t} \boldsymbol {\beta} _ {i, n} ^ {t} ^ {\top} \right\| _ {2}\right) \right] \tag {22} \\ \leq C _ {x} ^ {2} d _ {s} \left[ \left\| \boldsymbol {\theta} ^ {t ^ {\prime}} - \boldsymbol {\theta} ^ {t} \right\| _ {2} + \bar {\lambda} _ {K} \left(\left\| \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} - \boldsymbol {\beta} _ {i, n} ^ {t} \right\| _ {\infty} + \left(\left\| \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} \right\| _ {2} + \left\| \boldsymbol {\beta} _ {i, n} ^ {t} \right\| _ {2}\right) \left\| \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} - \boldsymbol {\beta} _ {i, n} ^ {t} \right\| _ {2}\right) \right] \\ \leq C _ {x} ^ {2} d _ {s} \left[ \left\| \boldsymbol {\theta} ^ {t ^ {\prime}} - \boldsymbol {\theta} ^ {t} \right\| _ {2} + 3 \bar {\lambda} _ {K} \left\| \boldsymbol {\beta} _ {i, n} ^ {t ^ {\prime}} - \boldsymbol {\beta} _ {i, n} ^ {t} \right\| _ {2} \right] \\ \leq C _ {x} ^ {2} d _ {s} \left[ \left\| \boldsymbol {\theta} ^ {t ^ {\prime}} - \boldsymbol {\theta} ^ {t} \right\| _ {2} + 3 \bar {\lambda} _ {K} 2 \tau_ {0} C _ {x} ^ {2} d _ {s} \left(\bar {\lambda} _ {Q} \left\| \boldsymbol {W} _ {K} ^ {t ^ {\prime}} - \boldsymbol {W} _ {K} ^ {t} \right\| _ {2} + \bar {\lambda} _ {K} \left\| \boldsymbol {W} _ {Q} ^ {t ^ {\prime}} - \boldsymbol {W} _ {Q} ^ {t} \right\| _ {2}\right) \right] \\ \leq \left[ C _ {x} ^ {2} d _ {s} + 3 \bar {\lambda} _ {K} z \right] \left\| \boldsymbol {\theta} ^ {t ^ {\prime}} - \boldsymbol {\theta} ^ {t} \right\| _ {2}, \\ \end{array}
$$