Lately I've seen some advantages mostly in model generalization of minimizing an the mean absolute error (or I guess Laplacian MLE would be an equivalent way of saying it). I'm debating first on what options I have to optimize with, and second of all some efficiency considerations for these options. I've made a list below of what I know so far as a starting point to get some feedback on feasibility/efficiency of these methods for this task. If anyone has some solid theory to inform on the advantages/disadvantages of these and any other methods, it would be greatly appreciated! Please consider model size as it grows larger/deeper as well...in that evaluating the model/running for a training iteration may be very expensive (so if a method facilitates analytical gradient-based optimization it would have the advantage of minimizing training time, all else equal).