Paper:
Automatic differentiation frameworks are optimized for exactly one thing: computing the average mini-batch gradient. Yet, other quantities such as the variance of the mini-batch gradients or many approximations to the Hessian can, in theory, be computed efficiently, and at the same time as the gradient. While these quantities are of great interest to researchers and practitioners, current deep-learning software does not support their automatic calculation. Manually implementing them is burdensome, inefficient if done naïvely, and the resulting code is rarely shared. This hampers progress in deep learning, and unnecessarily narrows research to focus on gradient descent and its variants; it also complicates replication studies and comparisons between newly developed methods that require those quantities, to the point of impossibility. To address this problem, we introduce BACKPACK 1 , an efficient framework built on top of PYTORCH, that extends the backpropagation algorithm to extract additional information from first-and second-order derivatives. Its capabilities are illustrated by benchmark reports for computing additional quantities on deep neural networks, and an example application by testing several recent curvature approximations for optimization.

Review:
This is a good paper. The authors present a software implementation which allows one to extend PyTorch to compute quantities that are irritatingly difficult to compute in PyTorch directly, or in other automatic differentiation frameworks, particularly if efficiency is a concern. Issues of this kind have been discussed at length within the community, particularly on GitHub, and related issues with optimization of automatic differentiation code have motivated other software developments, such as Julia's Zygote package. Having wasted a large amount of my time implementing the WGAN gradient penalty from scratch - which, to implement scalably, requires one to use both forward-mode and reverse-mode automatic differentiation simultaneously - I appreciate what the authors are trying to do here to make research that requires more sophisticated automatic differentiation more accessible. 
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Detailed remarks below.
* The paper's title is not very informative about what the paper is about. The authors should choose a different title - perhaps something like "BackPACK: user-friendly higher-order automatic differentiation in deep networks" or something similar but less long.
* The authors focus on KFAC-type methods as their key illustrated use case, but actually software frameworks like this are also helpful for certain GAN losses - WGAN-GP in particular. These losses require one to compute a term that involves the norm of a certain gradient of the output. The gradient of this gradient can be handled efficiently with Hessian-Vector products, which in principle are computable efficiently via automatic differentiation, but in practice a huge pain because of the need to use both forward-mode and reverse-mode automatic differentiation and lack of first-class support from automatic differentiation packages. Provided I've not misunderstood BackPACK and that it would help in making such computations less tedious (and I can't verify this myself, as my own implementation of WGAN-GP was not in PyTorch), I would highly recommend the authors to add an extra paragraph discussing this particular use case, because this would increase the paper's impact on the community by connecting it to another literature which is not mentioned in the paper.
* The entire paper could do with talking about backpropagation less, and automatic differentiation more, because it illustrates that the concerns addressed are not solely limited to deep networks, even if the package does primarily target them.
* P2: these issues are not limited to the Python community, and specialized automatic differentiation software has also been developed for Julia. The authors should cite Innes [1,2] and related papers from that literature.
* Figure 1: from the sample code, I worry about how generic BackPACK is. I think the package authors should be careful not to specialize too much to particular kinds of deep networks, particularly since a much wider variety of models and network architectures are starting to be used with automatic differentiation.
* P2: capital \Omega notation is confusing, please replace with capital \Theta.
* P2: L_2 regularization should more technically be \ell^2 instead.
* P4: please cite Baydin [3] who provides a very nice review of automatic differentiation. It may help explanation to introduce dual numbers, which make forward-mode much easier to understand.
* P6: please write out "with respect to" for "w.r.t.".
* P7: I really liked this section. The simplicity of implementing the example method using the authors' software framework feels compelling to me. However, genericness is still a concern: by analogy, every deep learning framework can do MNIST easily, but some of them make it much harder to do customized or advanced implementation than others. The latter cases are often the ones that matter to practitioners. It's hard to tell how easy it will be to implement something the authors did not foresee or consider - but this will necessarily be the case in any software paper.
* P8: "and in part driven" - missing a comma.
* P8: please spell out "Table" in "Tab. 1".
[1] M. Innes. Don't Unroll Adjoint: Differentiating SSA-Form Programs. NeurIPS, 2018.
[2] M. Innes. Flux: Elegant Machine Learning with Julia. Journal of Open Source Software, 2018.
[3] Baydin, Atilim Gunes and Pearlmutter, Barak A and Radul, Alexey Andreyevich and Siskind, Jeffrey Mark. Automatic differentiation in machine learning: a survey. JMLR, 2017.