 \sum^T_{t=0}\left(\eta^{\bd{\gamma},\bd{f}}_t -\eta^{P,\bd{f}}_t\right) + \sum^T_{t=0} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_0 \nonumber\\     &= \sum^T_{t=0}\left[\ave{\pi^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=0} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_0.