I understand that RMSE is just the square root of MSE. Generally, as far as I have seen, people seem to use MSE as a loss function and RMSE for evaluation purposes, since it exactly gives you the error as a distance in the Euclidean space.
What could be a major difference between using MSE and RMSE when used as loss functions for training?
I'm curious because good frameworks like PyTorch, Keras, etc. don't provide RMSE loss functions out of the box. Is it some kind of standard convention? If so, why?
Also, I'm aware of the difference that MSE magnifies the errors with magnitude>1 and shrinks the errors with magnitude<1 (on a quadratic scale), which RMSE doesn't do.