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\begin{array}{l} \mathbb {E} _ {\mathcal {D}} \big [ \| x _ {k + 1} - x ^ {*} \| ^ {2} + \| x _ {k + 1} - \hat {x} _ {k} \| ^ {2} \big ] \stackrel {(3 1)} {=} \| x _ {k} - x ^ {*} \| ^ {2} + 2 \mathbb {E} _ {\mathcal {D}} \| x _ {k + 1} - \hat {x} _ {k} \| ^ {2} - \| \hat {x} _ {k} - x _ {k} \| ^ {2} \\ - 2 \omega \mathbb {E} _ {\mathcal {D}} \left\langle F _ {v _ {k}} (\hat {x} _ {k}), \hat {x} _ {k} - x ^ {*} \right\rangle \\ = \left\| x _ {k} - x ^ {*} \right\| ^ {2} + 2 \mathbb {E} _ {\mathcal {D}} \left\| x _ {k + 1} - \hat {x} _ {k} \right\| ^ {2} - \left\| \hat {x} _ {k} - x _ {k} \right\| ^ {2} \\ - 2 \omega \left\langle F (\hat {x} _ {k}), \hat {x} _ {k} - x ^ {*} \right\rangle \\ \stackrel {(3)} {\leq} \left\| x _ {k} - x ^ {*} \right\| ^ {2} + 2 \mathbb {E} _ {\mathcal {D}} \left\| x _ {k + 1} - \hat {x} _ {k} \right\| ^ {2} - \left\| \hat {x} _ {k} - x _ {k} \right\| ^ {2} \\ - 2 \omega \mu \| \hat {x} _ {k} - x ^ {*} \| ^ {2} \\ = \left\| x _ {k} - x ^ {*} \right\| ^ {2} + 2 \omega^ {2} \mathbb {E} _ {\mathcal {D}} \left\| F _ {v _ {k}} \left(\hat {x} _ {k}\right) - F _ {v _ {k - 1}} \left(\hat {x} _ {k - 1}\right) \right\| ^ {2} \\ - \left\| \hat {x} _ {k} - x _ {k} \right\| ^ {2} - 2 \omega \mu \left\| \hat {x} _ {k} - x ^ {*} \right\| ^ {2} \\ \stackrel {(3 2)} {\leq} \| x _ {k} - x ^ {*} \| ^ {2} + 2 \omega^ {2} \left(\delta \| \hat {x} _ {k} - x ^ {*} \| ^ {2} + 2 \delta \| \hat {x} _ {k - 1} - x ^ {*} \| ^ {2} \right. \\ \left. + 2 L ^ {2} \left\| \hat {x} _ {k} - \hat {x} _ {k - 1} \right\| ^ {2} + 6 \sigma_ {*} ^ {2}\right) - \left\| \hat {x} _ {k} - x _ {k} \right\| ^ {2} \\ - 2 \omega \mu \| \hat {x} _ {k} - x ^ {*} \| ^ {2} \\ = \left\| x _ {k} - x ^ {*} \right\| ^ {2} - 2 \omega (\mu - \omega \delta) \left\| \hat {x} _ {k} - x ^ {*} \right\| ^ {2} \\ + 4 \omega^ {2} \delta \| \hat {x} _ {k - 1} - x ^ {*} \| ^ {2} + 4 \omega^ {2} L ^ {2} \| \hat {x} _ {k} - \hat {x} _ {k - 1} \| ^ {2} \\ - \left\| \hat {x} _ {k} - x _ {k} \right\| ^ {2} + 1 2 \omega^ {2} \sigma_ {*} ^ {2} \\ \stackrel {(2 2)} {\leq} \quad \| x _ {k} - x ^ {*} \| ^ {2} - \omega (\mu - \omega \delta) \| x _ {k} - x ^ {*} \| ^ {2} \\ + 2 \omega (\mu - \omega \delta) \| x _ {k} - \hat {x} _ {k} \| ^ {2} + 4 \omega^ {2} \delta \| \hat {x} _ {k - 1} - x ^ {*} \| ^ {2} \\ + 4 \omega^ {2} L ^ {2} \| \hat {x} _ {k} - \hat {x} _ {k - 1} \| ^ {2} - \| \hat {x} _ {k} - x _ {k} \| ^ {2} \\ + 1 2 \omega^ {2} \sigma_ {*} ^ {2} \\ \stackrel {(2 2)} {\leq} \quad \| x _ {k} - x ^ {*} \| ^ {2} - \omega (\mu - \omega \delta) \| x _ {k} - x ^ {*} \| ^ {2} \\ + 2 \omega (\mu - \omega \delta) \| x _ {k} - \hat {x} _ {k} \| ^ {2} + 8 \omega^ {2} \delta \| \hat {x} _ {k - 1} - x _ {k} \| ^ {2} \\ + 8 \omega^ {2} \delta \| x _ {k} - x ^ {*} \| ^ {2} + 8 \omega^ {2} L ^ {2} \| \hat {x} _ {k} - x _ {k} \| ^ {2} \\ + 8 \omega^ {2} L ^ {2} \| x _ {k} - \hat {x} _ {k - 1} \| ^ {2} - \| \hat {x} _ {k} - x _ {k} \| ^ {2} + 1 2 \omega^ {2} \sigma_ {*} ^ {2} \\ = (1 - \omega \mu + 9 \omega^ {2} \delta) \| x _ {k} - x ^ {*} \| ^ {2} \\ + \left(8 \omega^ {2} \delta + 8 \omega^ {2} L ^ {2}\right) \| x _ {k} - \hat {x} _ {k - 1} \| ^ {2} \\ + (2 \omega (\mu - \omega \delta) + 8 \omega^ {2} L ^ {2} - 1) \| x _ {k} - \hat {x} _ {k} \| ^ {2} + 1 2 \omega^ {2} \sigma_ {*} ^ {2}. \\ \end{array}
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