\begin{align}
    \nabla_w \ell(w_t,S)^T (w_{t+1}-w^* )  +\frac{1}{2\alpha_t} ||w_{t+1} - w_t||^2&=   \nabla_w \ell(w_t,S)^T (w_{t+1}-w_t ) +\frac{1}{2\alpha_t} ||w_{t+1} - w_t||^2 \nonumber \\ & ~~~~~~ - \nabla_w \ell(w_t,S)^T (w^*-w_t ) \nonumber  \\
    &\geq (\ell(w_{t+1}) - \ell(w_t)) - (\ell(w^*) - \ell(w_t)) \nonumber  \\ & = \ell(w_{t+1}) - \ell(w^*) \nonumber 
\end{align}
