\begin{equation}\label{eq:def:running_avg}
\widehat{w}_t(S_n) = \frac{\sum_{k=0}^t \alpha_k w_{k+1}(S_n)}{\sum_{k=0}^t \alpha_k}.
\end{equation} % Any minimizer $w^*(S_n)$ may be used in the above equation in the case of multiple minimizers. 
