On non-approximability of zero loss global $\mathcal{L}^2$ minimizers by gradient descent in deep learning

We analyze geometric aspects of the gradient descent algorithm in Deep Learning (DL), and give a detailed discussion of the circumstance that, in underparametrized DL networks, zero loss minimization cannot generically be attained. As a consequence, we conclude that the distribution of training inputs must necessarily be non-generic in order to produce zero loss minimizers, both for the method constructed in [2, 3], or for gradient descent [1] (which assume clustering of training data).