1. It is unclear how guaranteeing stationary points that have small gradient norms translates to good generalization. The bounds just indicate that these algorithms reach one of the many stationary points for adaptive gradient methods and don't talk about how reaching one of the potentially many population stationary points especially in the non-convex regime can translate to good generalization. A remark on this would be helpful. 2. Line 124-125: For any w, the Hoeffding's bound holds true as long as the samples are drawn independently and so it is always possible to show inequality (2). Stochastic algorithms moreover impose conditioning on the previous iterate further guaranteeing that Hoeffding inequality holds. It will be great if the authors can elaborate on this. 3. The bounds in Theorem 1 have a dependence on d, which the authors have discussed. However, if \mu is small, the bounds are moot. If \mu is large, then the concentration guarantees are not very useful. Based on values in Theorem 2, latter seems to be the case. 4. It seems weird that the bounds in Theorems 2 and 4 do not depend on the initialization w_0 but on w_1. 5. For experiments on Penn-Tree bank, it seems that the algorithms are not stable with respect to train perplexity.
2. Line 124-125: For any w, the Hoeffding's bound holds true as long as the samples are drawn independently and so it is always possible to show inequality (2). Stochastic algorithms moreover impose conditioning on the previous iterate further guaranteeing that Hoeffding inequality holds. It will be great if the authors can elaborate on this.