Paper:
Published as a conference paper at ICLR 2018 NEUMANN OPTIMIZER: A PRACTICAL OPTIMIZATION ALGORITHM FOR DEEP NEURAL NETWORKS
Progress in deep learning is slowed by the days or weeks it takes to train large models. The natural solution of using more hardware is limited by diminishing returns, and leads to inefficient use of additional resources. In this paper, we present a large batch, stochastic optimization algorithm that is both faster than widely used algorithms for fixed amounts of computation, and also scales up substantially better as more computational resources become available. Our algorithm implicitly computes the inverse Hessian of each mini-batch to produce descent directions; we do so without either an explicit approximation to the Hessian or Hessian-vector products. We demonstrate the effectiveness of our algorithm by successfully training large ImageNet models (Inception-V3, Resnet-50, Resnet-101 and Inception-Resnet-V2) with mini-batch sizes of up to 32000 with no loss in validation error relative to current baselines, and no increase in the total number of steps. At smaller mini-batch sizes, our optimizer improves the validation error in these models by 0.8-0.9%. Alternatively, we can trade off this accuracy to reduce the number of training steps needed by roughly 10-30%. Our work is practical and easily usable by others -only one hyperparameter (learning rate) needs tuning, and furthermore, the algorithm is as computationally cheap as the commonly used Adam optimizer.

Review:
This paper presents a new 2nd-order algorithm that implicitly uses curvature information, and it shows the intuition behind the approximation schemes in the algorithms and also validates the heuristics in various experiments.  The method involves using Neumann Series and Richardson iteration to avoid Hessian-vector product in second order method for NN.  In the actual performance, the paper presents both practical efficiency and better generalization error in different deep neural networks for image classification tasks, and the authors also show differences according to different settings, e.g., Batch Size, Regularization.  The numerical examples are relatively clear and easy to figure out details.
1. While the paper presents the algorithm as an optimization algorithm, although it gets better learning performance, it would be interesting to see how well it is as an optimizer.  For example, one simple experiment would be showing how it works for convex problems, e.g., logistic regression.  Realistic DNN systems are very complex, and evaluating the method in a simple setting would help a lot in determining what if anything is novel about the method.
2. Also, for deep learning problems, it would be more convincing to see how different initialization can affect the performances. 
3. Although the authors present their algorithm as a second order method at beginning, the final algorithm is kind of like a complex momentum SGD with limited memory.  Rather than simply throwing out a new method with a new name, it would be helpful to understand what the steps of this method are implicitly doing.  Please explain more about this.
4. It said that the algorithm is hyperparameter free except for learning rate.  However, it is hard to see why there is no need to tune other hyperparameters, e.g., Cubic Regularizer, Repulsive Regularizer.  The effect/sensitivity of hyperparameters for second order methods are quite different than hyperparameters for first order methods, and it is of interest to know how hyperparameters for implicit second order methods perform.
5. For Section 4.2, the well know benefit by using large batch size to train models is that it could reduce training time and epochs.  However, from Table 3, there is no such phenomenon.  Please explain.