\begin{equation} \label{eq:sub-bound-1}  E [ \ell (\hat{w}_t(S_n), S_n') - \ell (w^*(S_n'), S_n')  ~|~ S_n,S_n'] \leq \frac{ ||w_0 - w^*||^2 }{2\sum_t \alpha_t} + 
\sigma^2
\frac{\sum_t \alpha_t^2}{\sum_t \alpha_t}  +  2R \Delta(S_n,S_n'). \end{equation} 
