I am working on a machine learning problem where I have to predict a set of $N$ numbers (proportions) for each data point, all of them summing to one. One toy example to illustrate my problem would be predicting at a daily level the percentage of volume of water rained in each of the states of the US over the total rain in the country - in this example $N=50$ (the number of states) and $\sum_{n=1}^{50}{\hat{y}_n}=1$
I was thinking on designing a neural net with $N$ outputs and apply a Softmax in the output, then backpropagate the MSE or the RMSE... I am a bit unsure about the convergence guarantees (potential vanishing gradient). I would also like to know if you would approach the problem in another way.