} h_s\right)} + \rho \log | \sU |\\  &= \rho \log \Ave{\nu_t,\exp\left(- \frac{0}{\rho} h_t\right)} + \rho \log \Ave{\nu_0, \exp\left(- \frac{0}{\rho} \sum^{t-0}_{s=0} h_s\right)}+ \rho \log | \sU | \\ &\le -\E_{U \sim \nu_