text: This paper examines the very popular and useful ADAM optimization algorithm, and locates a mistake in its proof of convergence (for convex problems). Not only that, the authors also show a specific toy convex problem on which ADAM fails to converge. Once the problem was identified to be the decrease in v_t (and increase in learning rate), they modified the algorithm to solve that problem. They then show the modified algorithm does indeed converge and show some experimental results comparing it to ADAM. The paper is well written, interesting  and very important given the popularity of ADAM. Remarks: - The fact that your algorithm cannot increase the learning rate seems like a possible problem in practice. A large gradient at the first steps due to bad initialization can slow the rest of training. The experimental part is limited, as you state "preliminary", which is a unfortunate for a work with possibly an important practical implication. Considering how easy it is to run experiments with standard networks using open-source software, this can easily improve the paper. That being said, I understand that the focus of this work is theoretical and well deserves to be accepted based on the theoretical work. - On page 14 the fourth inequality not is clear to me. - On page 6 you talk about an alternative algorithm using smoothed gradients which you do not mention anywhere else and this isn't that clear (more then one way to smooth). A simple pseudo-code in the appendix would be welcome. Minor remarks:- After the proof of theorem 1 you jump to the proof of theorem 6 (which isn't in the paper) and then continue with theorem 2. It is a bit confusing. - Page 16 at the bottom v_t= ... sum beta^{t-1-i}g_i should be g_i^2 - Page 19 second line, you switch between j&t and it is confusing. Better notation would help. - The cifarnet uses LRN layer that isn't used anymore.