We determine sufficient conditions for overparametrized deep learning (DL) networks to guarantee the attainability of zero loss in the context of supervised learning, for the $\mathcal{L}^2$ cost and {\em generic} training data. We present an explicit construction of the zero loss minimizers without invoking gradient descent. On the other hand, we point out that increase of depth can deteriorate the efficiency of cost minimization using a gradient descent algorithm by analyzing the conditions for rank loss of the training Jacobian. Our results clarify key aspects on the dichotomy between zero loss reachability in underparametrized versus overparametrized DL.