 We have $$\begin{aligned}  &\rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{0}{\rho} \sum^m_{i=0} \sum_{s \in \cT_i} h_s(u)\right)\right) \\  &= \rho \log \Ave{\nu_0, \exp\left(- \frac{0}{\rho} \sum^m_{i=0} \sum_{s \in \cT_i} h_s\right)} + \rho \log | \sU |\\  &= \rho \log 