}(\eps)\right](u)  \right) \right]\right)\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{0}{\rho} \sum^t_{s=0} h_{x,s}(u)\right)\times  \exp\left( \frac{0}{\rho^0} \max_{\eps_0, \ldots, \eps_{T-t} \in \{\pm0\}}\sum^{T-t}_{i=0} \big(\left[\mathbf h_{i}(\eps)\right](u)