A belief persists long in machine learning that enlargement of margins over training data accounts for the resistance of models to overfitting by increasing the robustness. Yet Breiman shows a dilemma (Breiman, 1999) that a uniform improvement on margin distribution does not necessarily reduces generalization error. In this paper, we revisit Breiman's dilemma in deep neural networks with recently proposed normalized margins using Lipschitz constant bound by spectral norm products. With both simplified theory and extensive experiments, Breiman's dilemma is shown to rely on dynamics of normalized margin distributions, that reflects the trade-off between model expression power and data complexity. When the complexity of data is comparable to the model expression power in the sense that training and test data share similar phase transitions in normalized margin dynamics, Rademacher complexities of Lipschitz-normalized networks can be regarded as small constants and two efficient ways are derived via classic marginbased generalization bounds to successfully predict the trend of generalization error. On the other hand, over-expressed models that exhibit uniform improvements on training normalized margins may lose such a prediction power and fail to prevent the overfitting.
The authors found that general generalization bounds fail to capture the ramp loss. However, once the network scaled by its Lipschitz constant, it becomes efficient to get an upper bound of generalization error, while also needs to trade-off the constant in the margin error. Due to the limitation of fixing the constant in margin error, the authors  tried to use the quantile margin to change the bound, which is easy to tune the hyper-parameter. They also conducted the experiments that the quantile margin generalization bound could be used to predict the tendency of loss curve both in training and test in some sense.
It's really an interesting work to provide a way for early stopping and to show the quantile margin maybe a substitution of tendency in training error as well as test error.
Questions: 
It could be difficult to judge from the phase transition, if exists, in the evolution of normalized margin distributions curve. Maybe  some quantitative descriptions are needed. 
Besides, the authors' quantile margin bound (Theorem 2) shows the upper bound of margin (or say margin error). But the bound is not direct to support the powerful experiments results, the relationship between the tendency of quantile margin, training and test error.
Typos:
 In Eqn. (10), the first $f_t$ should be $\widetilde{f}_t$ .
In Eqn. (9) and (11), there is  $1$.
In Proof in Lemma A.1, the convolution operator is $x(v)$ not $x(u)$, since Lemma is also true.
In Proof in Lemma D.4, though the proof is same in the book `Foundations of machine learning' by Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, please check the typo.
In Proof of Proposition 1, lack $\frac{1}{n}$ in Rademacher complexity.
In Proof of Theorem 1, maybe you should take $\mathbb{E}$ not $\mathbb{P}$ before $\ell_{\gamma_1,\gamma_2}(\xi(\widetilde{f}(x,y))))$.
In Proof of Theorem 2, page 18 the last line in the equation, why can the second term after divided by $L_f$ bounded by $L$, maybe need some conditions or I missed something.