Backpropagation with a different sized training set?
Your cost function implies that you want the network to predict the magnitude of the output vector based on the magnitude of the input vector. Is that what you want to do? If not, e.g. if you want to accurately predict the output vector components, you need to believe that these are predictable based only on the magnitude of the inputs. If you don’t think this is possible, then you need to find new data. If you do, then a normal error function should be fine (e.g. MSE) and you can use normal backdrop as implemented in any NN library. If you’re getting poor performance, it’s probably because the norm of the input vector does not carry enough information to make a good prediction.