I have seen people speaking of the Log-Cosh Loss that is 2 times differentiable and mimic the Mean Absolute Error goers 0. It is therefore useful for algorithm that need hessian.
Hence I'm not sure to understand why MAE is not sufficient since it is not differentiable only in one point and this point is when our prediction is perfect.
However my intuition is that this log cosh loss can be pretty cool near 0 to reduce parameters update by decreasing the gradient.
Am I understanding this all right? Does anyone knows any specific example where this loss is used?