\begin{align} \label{eq:intialRegBoundUCB}
\begin{split}
    R_t &\leq \frac{2c_2}{c_1} \sum_{l=1}^L \left(\sqrt{\lambda} m_2 + \sqrt{2\log\left(\frac{1}{\delta}\right) + \log\left(\frac{\det\left(V_{t-1}^l(\lambda)\right)}{\lambda^d}\right)}\right) \min\left\{1\; , \; \|v_{a_t^l} + w_l v_{a_t^{l-1}}\|_{V_{t-1}^{l^{-1}}}\right\}
\end{split}
\end{align}
