The objective function is defined heuristically, and involves about five different sub-objectives (top of page four). Some of the parameters chosen seem to be rough guesses, as does the decision to scale up the images to twice the resolution when moving from classification (the pre-training task) to detection.

It seems miraculous that a process of estimating and refinement, guided by experience, can work on tasks where you have no mathematical guarantee that a good solution can be found. Maybe in time we'll build the theory that explains just why deep learning works so well, but for now I'm just kinda awed and impressed every time one of these stories comes out.