Paper:
ADAPTIVE GRADIENT METHODS WITH DYNAMIC BOUND OF LEARNING RATE
Adaptive optimization methods such as ADAGRAD, RMSPROP and ADAM have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGRAD to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of ADAM and AMSGRAD, called ADABOUND and AMSBOUND respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound.

Review:
The authors introduce AdaBound, a method that starts off as Adam but eventually transitions to SGD. The motivation is to benefit from the rapid training process of Adam in the beginning and the improved convergence of SGD at the end. The authors do so by clipping the weight updates of Adam in a dynamic way. They show numerical results and theoretical guarantees. The numerical results are presented on CIFAR-10 and PTB while the theoretical results are shown on assumptions similar to AMSGrad (& using similar proof strategies). As it stands, I have some foundational concerns about the paper and believe that it needs significant improvement before it can be published. I request the authors to please let me know if I misunderstood any aspect of the algorithm, I will adjust my rating promptly. I detail my key criticisms below:
- I'm somewhat confused by the formulation of \eta_u and \eta_l. The way it is set up (end of Section 4), the final learning rate for the algorithm converges to 0.1 as t goes to infinity. In the Appendix, the authors show results also with final convergence to 1. Are the results coincidental with the fact that SGD works well with those learning rates? It is a bit odd that we indirectly encode the final learning rate of the algorithm into the \eta s. 
- Am I correct in saying that with t=100 (i.e., the 100th iteration), the \eta s constrain the learning rates to be in a tight bound around 0.1? If beta=0.9, then \eta_l(1) = 0.1 - 0.1 / (0.1*100+1) = 0.091. After t=1000 iterations, \eta_l becomes 0.099. Again, are the good results coincidental with the fact that SGD with learning rate 0.1 works well for this setup? In the scheme of the 200 epochs of training (equaling almost 100-150k iterations), if \eta s are almost 0.099 / 0.10099, for over 99% of the training, we're only doing SGD with learning rate 0.1. 
- Along the same lines, what learning rates on the grid were chosen for each of the problems? Does the setup still work if SGD needs a small step size and we still have \eta converge to 1? A VGG-11 without batch normalization typically needs a smaller learning rate than usual; could you try the algorithms on that? 
- Can the authors plot the evolution of learning rate of the algorithm over time? You could pick the min/median/max of the learning rates and plot them against epochs in the same way as accuracy.This would be a good meta-result to show how gradual the transition from Adam to SGD is. 
 
- The core observation of extreme learning rates and the proposal of clipping the updates is not novel; Keskar and Socher (which the authors cite for other claims) motivates their setup with the same idea (Section 2 of their paper). I feel that the authors should clarify what they are proposing as novel. Is it correct that a careful theoretical analysis of this framework is what stands as the authors' major contribution?
- Can you try experimenting with/suggesting trajectories for \eta which converge to SGD stepsize more slower? 
- Similarly, can you suggest ways to automate the choice for the \eta^\star? It seems that the 0.1 in the numerator is an additional hyperparameter that still might need tuning?