$$
\begin{array}{l} + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta \left(\mathbb {E} \left[ \ell_ {t + \tau} \left(w _ {\text {r e f}}\right) \mid \mathcal {F} _ {t} \right] - \mathbb {E} \left[ \ell_ {t + \tau} \left(w _ {t}\right) \mid \mathcal {F} _ {t} \right] + \ell_ {t + \tau} \left(w _ {\text {r e f}}\right) - \ell_ {t + \tau} \left(w _ {t}\right)\right) \\ + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta \left(\ell_ {t + \tau} (w _ {t}) - \ell_ {t + \tau} (w _ {\text {r e f}})\right) + \sum_ {t = 0} ^ {k} z _ {t} \eta \left(\ell_ {t + 1} (w _ {\text {r e f}}) - \ell_ {t + 1} (w _ {t})\right) \\ + \sum_ {t = 0} ^ {k - \tau} \left(z _ {t} - z _ {t - 1}\right) \mathbf {D} _ {\psi} \left(w _ {\text {r e f}}; w _ {t}\right) + \sum_ {t = 0} ^ {k} \frac {z _ {t} \eta^ {2}}{2} \| g _ {t + 1} \| _ {*} ^ {2} \\ \leq \sum_ {t = 0} ^ {k} z _ {t} \eta \left(\mathcal {R} \left(w _ {\text {r e f}}\right) - \mathcal {R} ^ {*}\right) + \sum_ {t = k - \tau + 1} ^ {k} z _ {t} \eta \left(\mathcal {R} \left(w _ {t}\right) - \mathcal {R} \left(w _ {\text {r e f}}\right)\right) \\ + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta (\mathcal {R} (w _ {t}) - \mathcal {R} (w _ {\text {r e f}}) - \mathbb {E} [ \ell_ {t + \tau} (w _ {\text {r e f}}) \mid \mathcal {F} _ {t} ] + \mathbb {E} [ \ell_ {t + \tau} (w _ {t}) \mid \mathcal {F} _ {t} ]) \\ + \tau \log \frac {1}{\delta} + \sum_ {t = 0} ^ {k - \tau} 8 (z _ {t} \eta) ^ {2} C _ {4} ^ {2} \mathbf {D} _ {\psi} (w _ {\text {r e f}}; w _ {t}) + \sum_ {t = 0} ^ {k - \tau} (z _ {t} - z _ {t - 1}) \mathbf {D} _ {\psi} (w _ {\text {r e f}}; w _ {t}) \\ + \sum_ {t = 0} ^ {k} \frac {z _ {t} \eta^ {2}}{2} \| g _ {t + 1} \| _ {*} ^ {2} + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta (\ell_ {t + \tau} (w _ {t}) - \ell_ {t + \tau} (w _ {t + \tau - 1})) \\ + \sum_ {t = \tau - 1} ^ {k - 1} z _ {t - \tau + 1} \eta \left(\ell_ {t + 1} (w _ {t}) - \ell_ {t + 1} (w _ {\text {r e f}})\right) + \sum_ {t = 0} ^ {k} z _ {t} \eta \left(\ell_ {t + 1} (w _ {\text {r e f}}) - \ell_ {t + 1} (w _ {t})\right) \\ \leq \sum_ {t = 0} ^ {k} z _ {t} \eta \left(\mathcal {R} \left(w _ {\text {r e f}}\right) - \mathcal {R} ^ {*}\right) + \sum_ {t = k - \tau + 1} ^ {k} z _ {t} \eta \left(\mathcal {R} \left(w _ {t}\right) - \mathcal {R} \left(w _ {\text {r e f}}\right)\right) \\ + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta (\mathcal {R} (w _ {t}) - \mathcal {R} (w _ {\text {r e f}}) - \mathbb {E} [ \ell_ {t + \tau} (w _ {\text {r e f}}) \mid \mathcal {F} _ {t} ] + \mathbb {E} [ \ell_ {t + \tau} (w _ {t}) \mid \mathcal {F} _ {t} ]) + \tau \log \frac {1}{\delta} \\ + \sum_ {t = 0} ^ {k} \frac {z _ {t} \eta^ {2}}{2} \| g _ {t + 1} \| _ {*} ^ {2} + \sum_ {t = 0} ^ {k - \tau} z _ {t} \eta (\ell_ {t + \tau} (w _ {t}) - \ell_ {t + \tau} (w _ {t + \tau - 1})) \\ + \sum_ {t = \tau - 1} ^ {k - 1} \eta \left(z _ {t - \tau + 1} - z _ {t}\right) \left(\ell_ {t + 1} \left(w _ {t}\right) - \ell_ {t + 1} \left(w _ {\text {r e f}}\right)\right) \\ + \sum_ {t = 0} ^ {\tau - 2} z _ {t} \eta \left(\ell_ {t + 1} \left(w _ {\text {r e f}}\right) - \ell_ {t + 1} \left(w _ {t}\right)\right) + z _ {k} \eta \left(\ell_ {k + 1} \left(w _ {\text {r e f}}\right) - \ell_ {k + 1} \left(w _ {k}\right)\right) \\ \end{array}
$$