Paper:
A SCALABLE LAPLACE APPROXIMATION FOR NEURAL NETWORKS
We leverage recent insights from second-order optimisation for neural networks to construct a Kronecker factored Laplace approximation to the posterior over the weights of a trained network. Our approximation requires no modification of the training procedure, enabling practitioners to estimate the uncertainty of their models currently used in production without having to retrain them. We extensively compare our method to using Dropout and a diagonal Laplace approximation for estimating the uncertainty of a network. We demonstrate that our Kronecker factored method leads to better uncertainty estimates on out-of-distribution data and is more robust to simple adversarial attacks. Our approach only requires calculating two square curvature factor matrices for each layer. Their size is equal to the respective square of the input and output size of the layer, making the method efficient both computationally and in terms of memory usage. We illustrate its scalability by applying it to a state-of-the-art convolutional network architecture.

Review:
This paper uses recent progress in the understanding and approximation of curvature matrices in neural networks to revisit a venerable area- that of Laplace approximations to neural network posteriors. The Laplace method requires two stages - 1) obtaining a point estimate of the parameters followed by 2) estimation of the curvature. Since 1) is close to common practice it raises the appealing possibility of adding 2) after the fact, although the prior may be difficult to interpret in this case. A pitfall is that the method needs the point estimate to fall in a locally quadratic bowl or to add regularisation to make this true. The necessary amount of regularisation can be large as reported in section 5.4.
The paper is generally well written. In particular the mathematical exposition attains good clarity. Much of the mathematical treatment of the curvature was already discussed by Martens and Grosse and Botev et al in previous works. The paper is generally well referenced. 
Given the complexity of the method, I think it would have helped to submit the code in anonymized form at this point.There are also some experiments not there that would improve the contribution. Figure 1 should include a comparison to Hamiltonian Monte Carlo and the full Laplace approximation (It is not sufficient to point to experiments in Hernandez-Lobato and Adams 2015 with a different model/prior). The size of model and data would not be prohibitive for either of these methods in this instance. All that figure 1 shows at the moment is that the proposed approximation has smaller predictive variance than the fully diagonal variant of the method. 
It would be interesting (but perhaps not essential) to compare the Laplace approximation to other scalable methods from the literature such as that of Louizos and Welling 2016 which uses also used matrix normal distributions. It is good that the paper includes a modern architecture with a more challenging dataset. It is a shame the method does not work better in this instance but the authors should not be penalized for reporting this. I think a paper on a probabilistic method should at some point evaluate log likelihood in a case where the test distribution is the same as the training distribution. This complements experiments where there is dataset shift and we wish to show robustness. I would be very interested to know how useful the implied marginal likelihoods of the approximation where, as suggested for further work in the conclusion.