Paper:
The recent success of neural networks for solving difficult decision tasks has incentivized incorporating smart decision making "at the edge." However, this work has traditionally focused on neural network inference, rather than training, due to memory and compute limitations, especially in emerging non-volatile memory systems, where writes are energetically costly and reduce lifespan. Yet, the ability to train at the edge is becoming increasingly important as it enables real-time adaptability to device drift and environmental variation, user customization, and federated learning across devices. In this work, we address two key challenges for training on edge devices with non-volatile memory: low write density and low auxiliary memory. We present a low-rank training scheme that addresses these challenges while maintaining computational efficiency. We then demonstrate the technique on a representative convolutional neural network across several adaptation problems, where it out-performs standard SGD both in accuracy and in number of weight writes.

Review:
While inference on edge devices is a popular and well-studied problem in recent days, training on these devices comes with many challenges. This paper proposes a low-rank training schema that helps mitigate some of the critical challenges that occur during training models on NVM memory-based edge devices. Additionally, two techniques, namely streaming batch norm and gradient max norm, are proposed to help training in an online setting. The proposed method is mainly based on approximating the Kronecker sum and is largely inspired by (Benzing et al, ICML 2019, Optimal Kronecker-sum approximation of real time recurrent learning). The proposed approach provides a few optimizations that improves this performance further, and outperforms SGD in terms of accuracy and the number of weights updates in a limited experimental setting.
+ves:
+ Training on an edge device is a relevant setting, where there is very little work so far, and this is a useful objective.
+ Focusing on the Kronecker sum and speeding it seems like an interesting solution to solving this problem.
+ The experimental settings considered (e.g. flipping bits and adding Gaussian noise to weights) is interesting, and perhaps of larger relevance to other work in the area.
+ The ablation studies provided in the appendix are useful and appreciable.
Concerns:
- The key concern is that most of the proposed method is built upon Benzing et al’s work (ICML 2019), and the original contribution seems limited. While the paper introduces a few optimizations further, this seems to be incremental than originally novel.
- The problem is formulated for linear regression with least-squares loss, and the experimentation is carried out on the MNIST dataset (classification setting). How is the methodology relevant to the cross-entropy loss, typically used for classification? The paper does not talk about this.
- The results are shown for a network with four 3 x 3 convolution layers and two fully connected layers on the MNIST dataset. Results with different architectures on a few other datasets (at least CIFAR-10) would be necessary to assess the usefulness of this method. Some discussion on what would be the maximum depth of the network that can be trained using this training schema and hardware would be very useful. 
- Do all deep neural network architectures (and loss functions) admit a Kronecker sum representation? What class of models can benefit from this method? Factorizations such as Cholesky allow to interpolate between computational complexity and decomposition rank by tuning a rank hyperparameter. Why would such factorizations not be better for memory-constrained settings?
- It would be interesting to see the results using the proposed algorithm on standard hardware, will it provide the same performance as SGD when the scale of the problem increases in terms of layers and other regularization techniques, etc.? This would help understand the performance of the algorithm in a standard-setting.
Minor issues:
- The paper could have been organized better. Most of the main paper is used to describe Benzing et al’s method, and a lot of the details of the original contributions are in the Appendix. While the equations discussed in Section 4 hold for Linear Regression, the same cannot be directly extended for other networks like CNNs. Considering all the experiments are on CNNs, Sec B.2 in the Appendix should have been in the main paper to help a reader follow the experiments. 
=====POST-REBUTTAL COMMENTS========
I thank the authors for the rebuttal. The paper's organization is much better now. Unfortunately, there are many concerns which are still not convincingly answered: (i) Considering the focus of this work (as admitted by the authors in the rebuttal) is empirical and with limited technical novelty, more comprehensive results on many datasets are required. To some extent, transfer to ImageNet addresses this partially, but not convincingly enough; (ii) It is not clear from the authors' rebuttal why testing this on standard hardware is not possible/appropriate. The rebuttal says "hard to quantify due to a variety of factors" but that is vague without stating the factors.
I am willing to increase the rating from 3 to 4, but am unfortunately still towards rejecting the current shape of this work. I believe the work will benefit from considering all reviewers' comments more comprehensively to have a more impactful paper.