this proposal is larger than $0.8$, we set it as a foreground sample. If is lower than $0.3$, it will be set as background sample. The ratio between foreground and background samples is set as 1.0 during training. The localization loss is devised as Smooth L1 loss [@Girshick:ICCV15] () between the predicted foreground proposal and the closest ground truth instance of the proposal, which is computed by $$\label{Eq2:7}
       \small
       {L}_{loc} = S_{L1}(\varphi_{c}-g_{c})+S_{L1}(\varphi_{w}-g_{w}),$$ where and represents the center location and width of the proposal’s closest ground truth instance, respectively. For overlap loss, we adopt the mean square error (MSE) loss to optimize it as follows: $$\label{Eq2:8}
\small
L_{ov} = (y_{ov} - g_{iou})^2.$$

Eventually, the whole network is trained in an end-to-end manner by penalizing the three losses.

Prediction and Post-processing
------------------------------

During prediction of action localization, the final ranking score $y_f$ of each candidate action proposal depends on both action classification scores $\mathbf{y^a}$ and overlap parameter $y_{ov}$: $$\label{Eq2:10}
       \small
       y_f = \max(\mathbf{y^a}) \cdot y_{ov}.$$ Given the predicted action instance $\phi=\{\varphi_{c},\varphi_{w},C_a,y_f\}$ with refined boundary ($\varphi_{c},\varphi_{w}$), predicted action label $C_a$, and ranking score $y_f$, we employ the soft non-maximum suppression (soft-NMS) [@Bodla:ICCV17] for post-processing. In each iteration of soft-NMS, we represent the action 