\ave{\nu_t,f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=0} \ave{\nu,f_t} + \sum^T_{t=0} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_0,\end{aligned}$$ where in the first equality we have used 