text: This paper studies the question: Why does SGD on deep network is often successful, despite the fact that the objective induces bad local minima? The approach in this paper is to study a standard MNN with one hidden layer.  They show that in an overparametrized regime, where the number of parameters is logarithmically larger than the number of parameters in the input, the ratio between the number of (bad) local minima to the number of global minima decays exponentially.  They show this for a piecewise linear activation function, and input drawn from a standard Normal distribution.  Their improvement over previous work is that the required overparameterization is fairly moderate, and that the network that they considered is similar to ones used in practice.  This result seems interesting,  although it is clearly not sufficient to explain even the success on the setting studied in this paper,  since the number of minima of a certain type does not correspond to the probability of the SGD ending in one:  to estimate the latter, the size of each basin of attraction should be taken into account.  The authors are aware of this point and mention it as a disadvantage.  However, since this question in general is a difficult one,  any progress might be considered interesting.  Hopefully, in future work it would be possible to also bound the probability of starting in one of the basins of attraction of bad local minima. The paper is well written and well presented,  and the limitations of the approach, as well as its advantages over previous work, are clearly explained.  As I am not an expert on the previous works in this field, my judgment relies mostly on this work and its representation of previous work.  I did not verify the proofs in the appendix.