As of http://scikit-learn.org/stable/auto_examples/model_selection/plot_roc_crossval.html
The “steepness” of ROC curves is also important, since it is ideal to maximize the true positive rate while minimizing the false positive rate.
Others prefer to optimize the false positive rate by looking at a bounded AUC.
What are the advantages and disadvantages of each approach for maximizing fpr?
I believe that looking at the bounded AUC or at the steepness of the ROC curve is equivalent.
For example, in medical diagnosis, you want a steep ROC curve because you want the False Positive Rate (FPR) to be very low for a reasonably high True Positive Rate (TPR), e.g. you don't want to proceed with treatment for healthy patients.
In this case, if you were to choose a model with a steep ROC curve it would be equivalent to choosing a model with a high AUC in a bounded region near the origin.
Assuming that the ROC curve in the bounded region is approximately a line ($y=m x + b$), the area under the curve would be proportional to the slope of the line ($area=m$). Therefore the model with the steepest ROC curve and the model with the highest bounded AUC would be the same model.
