$$
\begin{array}{l} = 4 \mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {k} - \boldsymbol {\theta} _ {k + 1} \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ = 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ = 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) - h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) + h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] \\ = 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] + 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) - h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] \\ \stackrel {(d _ {4})} {=} 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) - h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, y _ {0} (\boldsymbol {\theta} _ {k})\right) + h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, y _ {0} (\boldsymbol {\theta} _ {k})\right) \right. \right. \\ \left. - h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) + h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) - H _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) \right\rVert^ {2} \left| \mathcal {F} _ {k - \tau} \right] \\ + 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) - h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] \\ \stackrel {(d _ {5})} {\leq} 2 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}\right) - h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, y _ {0} \left(\boldsymbol {\theta} _ {k}\right)\right) \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ + 2 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {k}, y _ {0} (\boldsymbol {\theta} _ {k})\right) - h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ + 2 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h _ {0} \left(X _ {k}, \boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) - H _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}, y _ {0} \left(\boldsymbol {\theta} _ {0} ^ {*}\right)\right) \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ + 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) - h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] \\ \stackrel {(d _ {6})} {\leq} 2 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| \hat {\lambda} _ {k} \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] + 6 0 0 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| \hat {\boldsymbol {\theta}} _ {k} \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ + 8 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| h (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) - h _ {0} (X _ {k}, \boldsymbol {\theta} _ {k}, \lambda_ {k}) \right\| ^ {2} | \mathcal {F} _ {k - \tau} \right] \\ \stackrel {(d _ {7})} {\leq} 2 4 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| \hat {\lambda} _ {k} \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] + 6 0 0 \alpha_ {k} ^ {2} \mathbb {E} \left[ \left\| \hat {\boldsymbol {\theta}} _ {k} \right\| ^ {2} \mid \mathcal {F} _ {k - \tau} \right] \\ + 8 \alpha_ {k} ^ {2} \mathcal {O} \left(c _ {1} ^ {3} \left(\| \boldsymbol {\theta} _ {0} \| + | \lambda_ {0} | + 1\right) ^ {3} \cdot m ^ {- 1 / 2}\right) \tag {42} \\ \end{array}
$$