$$
\begin{array}{l} T _ {5} = - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \langle \nabla f (z _ {t}) - f (x _ {t}), \alpha_ {t} \frac {g _ {t}}{b _ {t}} \rangle \right] \\ \stackrel {(i)} {\leq} \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| f (z _ {t}) - f (x _ {t}) \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ \stackrel {(i i)} {\leq} \frac {L ^ {2}}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| z _ {t} - x _ {t} \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ \stackrel {(i i i)} {=} \frac {L ^ {2}}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left\| \frac {\beta}{1 - \beta} (x _ {t} - x _ {t - 1}) + \frac {\alpha_ {t - 1}}{1 - \beta} (\delta_ {t - 1} + \beta e _ {t - 1} - e _ {t}) \right\| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ \stackrel {(i v)} {\leq} \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| x _ {t} - x _ {t - 1} \| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} \delta_ {t - 1} \| ^ {2} \right] \\ + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t - 1} \| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t} \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ \stackrel {(v)} {=} \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \left\| \left(\frac {m _ {t - 1}}{b _ {t - 1}} + \delta_ {t - 1} + e _ {t - 1} - e _ {t}\right) \right\| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} \delta_ {t - 1} \| ^ {2} \right] \\ + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t - 1} \| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t} \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ = \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left\| \frac {\alpha_ {t - 1} m _ {t - 1}}{b _ {t - 1}} \right\| ^ {2} \right] + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} \delta_ {t - 1} \| ^ {2} \right] \\ + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t - 1} \| ^ {2} \right] + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t} \| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} \delta_ {t - 1} \| ^ {2} \right] \\ + \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t - 1} \| ^ {2} \right] + \frac {L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| \alpha_ {t - 1} e _ {t} \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \left\| \frac {g _ {t}}{b _ {t}} \right\| ^ {2} \right] \\ = \frac {\beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \left\| \frac {m _ {t - 1}}{b _ {t - 1}} \right\| ^ {2} \right] + \frac {(1 + \beta^ {2}) L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \| \delta_ {t - 1} \| ^ {2} \right] \\ + \frac {2 \beta^ {2} L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \| e _ {t - 1} \| ^ {2} \right] + \frac {(1 + \beta^ {2}) L ^ {2}}{(1 - \beta) ^ {2}} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \| e _ {t} \| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} \| \frac {g _ {t}}{b _ {t}} \| ^ {2} \right] \\ \stackrel {(v i)} {\leq} \left(\frac {\beta^ {2} L ^ {2} d}{(1 - \beta) ^ {2}} + \frac {4 (1 + \beta^ {2}) L ^ {2} d}{(1 - \beta) ^ {2}} + \frac {2 \beta^ {2} L ^ {2} d}{(1 - \beta) ^ {2}} + \frac {(1 + \beta^ {2}) L ^ {2} d}{(1 - \beta) ^ {2}} + \frac {G ^ {2}}{2 \rho^ {2}}\right) \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \\ = \left(\frac {\left(8 \beta^ {2} + 1 0 \beta + 5\right) L ^ {2} d}{(1 - \beta) ^ {2}} + \frac {G ^ {2}}{2 \rho^ {2}}\right) \sum_ {t = 1} ^ {T} \alpha_ {t - 1} ^ {2} \tag {34} \\ \end{array}
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