Ok, so this may be a silly question, I know the literature recommends at length the simplex or interior point method to optimize a quantile regression problem. However I cannot figure out why a simple (stochastic or not) gradient descent isn't able to minimize the quantile loss which as a reminder looks like this.
Of course the gradient isn't defined at 0, but we can set it to 0 and in real life on large datasets with numerous dimensions, especially considering Doubles are rarely precisely equal with enough precision, it seems like a non issue.
I decided to experiment on a dataset and indeed it doesn't seem to work... I just can't figure out an intuition as to why. Can anybody help?
Follow up question, does anyone know of a way/library to fit a quantile regression with an easily distributable algorithm?