This work identifies a mistake in the existing proof of convergence of Adam,  which is among the most popular optimization methods in deep learning.  Moreover, it gives a simple 1-dimensional counterexample with linear losses on which Adam does not converge.  The same issue also affects RMSprop,  which may be viewed as a special case of Adam without momentum.  The problem with Adam is that the "learning rate" matrices V_t^{1/2}/alpha_t are not monotonically decreasing.  A new method, called AMSGrad is therefore proposed, which modifies Adam by forcing these matrices to be decreasing.  It is then shown that AMSGrad does satisfy essentially the same convergence bound as the one previously claimed for Adam.  Experiments and simulations are provided that support the theoretical analysis. Apart from some issues with the technical presentation (see below),  the paper is well-written. Given the popularity of Adam, I consider this paper to make a very interesting observation.  I further believe all issues with the technical presentation can be readily addressed. Issues with Technical Presentation:- All theorems should explicitly state the conditions they require instead of referring to "all the conditions in (Kingma & Ba, 2015)". - Theorem 2 is a repetition of Theorem 1 (except for additional conditions). - The proof of Theorem 3 assumes there are no projections,  so this should be stated as part of its conditions.  (The claim in footnote 2 that they can be handled seems highly plausible,  but you should be up front about the limitations of your results.) - The regret bound Theorem 4 establishes convergence of the optimization method,  so it plays the role of a sanity check.  However, it is strictly worse than the regret bound O(sqrt{T}) for online gradient descent [Zinkevich,2003],  so it cannot explain why the proposed AMSgrad method might be adaptive.  (The method may indeed be adaptive in some sense;  I am just saying the *bound* does not express that. This is also not a criticism of the current paper;  the same remark also applies to the previously claimed regret bound for Adam.) - The discussion following Corollary 1 suggests that sum_i hat{v}_{T,i}^{1/2} might be much smaller than d G_infty.  This is true,  but we should always expect it to be at least a constant,  because hat{v}_{t,i} is monotonically increasing by definition of the algorithm,  so the bound does not get better than O(sqrt(T)). It is also suggested that sum_i ||g_{1:T,i}|| = sqrt{sum_{t=1}^T g_{t,i}^2} might be much smaller than dG_infty,  but this is very unlikely,  because this term will typically grow like O(sqrt{T}), unless the data are extremely sparse,  so we should at least expect some dependence on T. - In the proof of Theorem 1, the initial point is taken to be x_1 = 1, which is perfectly fine,  but it is not "without loss of generality", as claimed.  This should be stated in the statement of the Theorem. - The proof of Theorem 6 in appendix B only covers epsilon=1.  If it is "easy to show" that the same construction also works for other epsilon, as claimed, then please provide the proof for general epsilon. Other remarks:- Theoretically, nonconvergence of Adam seems a severe problem.  Can you speculate on why this issue has not prevented its widespread adoption? Which factors might mitigate the issue in practice? - Please define g_t \circ g_t and g_{1:T,i} - I would recommend sticking with standard linear algebra notation for the sqrt and the inverse of a matrix and simply using A^{-1} and A^{1/2} instead of 1/A and sqrt{A}. - In theorems 1,2,3, I would recommend stating the dimension (d=1) of your counterexamples,  which makes them very nice! Minor issues:- Check accent on Nicol\`o Cesa-Bianchi in bibliography. - Near the end of the proof of Theorem 6: I believe you mean Adam suffers a "regret" instead of a "loss" of at least 2C-4. Also 2C-4=2C-4 is trivial in the second but last display.