Paper:
In modern supervised learning, many deep neural networks are able to interpolate the data: the empirical loss can be driven to near zero on all samples simultaneously. In this work, we explicitly exploit this interpolation property for the design of a new optimization algorithm for deep learning. Specifically, we use it to compute an adaptive learning-rate in closed form at each iteration. This results in the Adaptive Learning-rates for Interpolation with Gradients (ALI-G) algorithm. ALI-G retains the main advantage of SGD which is a low computational cost per iteration. But unlike SGD, the learning-rate of ALI-G uses a single constant hyper-parameter and does not require a decay schedule, which makes it considerably easier to tune. We provide convergence guarantees of ALI-G in the stochastic convex setting. Notably, all our convergence results tackle the realistic case where the interpolation property is satisfied up to some tolerance. We provide experiments on a variety of architectures and tasks: (i) learning a differentiable neural computer; (ii) training a wide residual network on the SVHN data set; (iii) training a Bi-LSTM on the SNLI data set; and (iv) training wide residual networks and densely connected networks on the CIFAR data sets. ALI-G produces state-of-the-art results among adaptive methods, and even yields comparable performance with SGD, which requires manually tuned learning-rate schedules. Furthermore, ALI-G is simple to implement in any standard deep learning framework and can be used as a drop-in replacement in existing code.

Review:
This paper proposes a new gradient descent methods for training deep neural network which can take the adaptive step size with only one hyper-parameter to tune -- the maximum learning rate -- and achieve comparable results to stochastic gradient descent (SGD) on various tasks and models. In order to achieve that, they develop a stochastic extension of the Polyak step-size for the non-convex setting, namely the adaptive learning-rates for interpolation with gradients (ALI-G), in which the minimal value of the objective loss is set to 0 due to interpolation in neural networks and the learning rates are clipped by a chosen maximal value. The problem is formulated clearly, and the review on the Polyak step-size and related works are well done. Another main contribution of the paper is to provide the convergence guarantees for ALI-G in the convex setting where the objective loss is Lipschitz-continuous (Theorem 1 in the paper). Their theorem also takes into account the error in the estimate of the minimal value of the objective loss. In addition, they derive the connections between  ALI-G and SGD and show that compared to SGD, ALI-G take into consideration that the objective loss is non-negative and set the loss to 0 when it is negative. They perform empirical study to compare their algorithm with other methods including Adagrad, Adam, DFW, L4Adam and SGD on learning a differentiable neural computer, object recognition, and a natural language processing task. Their experimental results show that ALI-G performance is comparable with that of SGD with schedule learning rate.
Overall, this paper could be an interesting contribution. However, some points in the theory and experiments need to be verified. I weakly reject this paper, but given these clarifications in an author response, I would be willing to increase the score. 
For the algorithm and theory, there are some points that need to be verified and further clarification on novelty:
1. When there are regularization such as the weight decay regularization, the minimal objective loss will not be 0. In such cases, Theorem 1 in the paper only guarantees that ALI-G reaches an objective loss less than or equal to a multiple of the estimate error \epsilon of the true minimal objective loss. It cannot guarantee that the objective loss reached by ALI-G can converge to the true minimall objective loss. Furthermore, when training neural networks with, for example, the weight decay regularization, often times the value of the regularization loss, i.e.  the estimate error \epsilon, is not small. Therefore, the upper bound given by Theorem 1 is rather loose. 
2.  The paper mentions that when no regularization is used, ALI-G and Deep Frank-Wolfe (DFW) are identical algorithms. The difference between the two algorithms are when regularization is used. However, given my concern for Theorem 1 above, the convergence of ALI-G and advantage of ALI-G over DFW in this setting is questionable, and the claim that “ALI-G can handle arbitrary (lower-bounded) loss functions” also needs to be verified. 
For the experiments, the following should be addressed:
1. In the experiment with the differentiable neural computers, even though ALI-G obtains better performance for a large range of \eta, its best objective loss is still worse than RMSProp, L4Adam, and L4Mom.
2. Given the merit of Theorem 1 is that the convergence guarantee takes into account the estimate error of the minimal objective loss, an ablation study that compares ALI-G with other methods in the same setting with and without regularization are needed. For example, it would be more convincing if similar results to those in Table 2 or 3 but without regularization are provided and discussed.
3. In Section 5.5, given that ALI-G and DFW are related, why is there no result for DFW in Figure 
4.?
4.  As the paper mentions, AProx algorithm and ALI-G are related, why is there no comparison with AProx in the experiments?
Things to improve the paper that did not impact the score:
1. In all experiments, the performance differences between ALI-G and competitive methods are small. Thus, error bars are needed for these results.
2. In Table 3, the gap between SGD and ALI-G can be significantly different on different architectures. For example, in CIFAR-100 experiments, while ALI-G achieves the same result as SGD with the DenseNet, ALI-G’s performance on Wide ResNet is much worse than SGD. Do you have any explanation for this?
3. How is ALI-G compared with the methods proposed in the paper “Stochastic Gradient Descent with Polyak's Learning Rate” (https://arxiv.org/abs/1903.08688)
The following paper also proved the convergence of adaptive gradient methods for nonconvex optimization:
Dongruo Zhou*, Yiqi Tang*, Ziyan Yang*, Yuan Cao, Quanquan Gu. On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization.
-------------------------------------------------------------
After rebuttal:
I noticed that the authors:
1. did not compare with SGD or Adam with good hyperparameters (https://arxiv.org/abs/1907.08610)
2. did not test on the large scale datasets, e.g., imagenet
3. the proposed algorithm is not as good as SGD on most numerical experiments.
For the above reasons, I think this paper should be rejected.