Published as a conference paper at ICLR 2018 GENERALIZING HAMILTONIAN MONTE CARLO WITH NEURAL NETWORKS
We present a general-purpose method to train Markov chain Monte Carlo kernels, parameterized by deep neural networks, that converge and mix quickly to their target distribution. Our method generalizes Hamiltonian Monte Carlo and is trained to maximize expected squared jumped distance, a proxy for mixing speed. We demonstrate large empirical gains on a collection of simple but challenging distributions, for instance achieving a 106× improvement in effective sample size in one case, and mixing when standard HMC makes no measurable progress in a second. Finally, we show quantitative and qualitative gains on a real-world task: latent-variable generative modeling. We release an open source TensorFlow implementation of the algorithm.
The paper introduces a non-volume-preserving generalization of HMC whose transitions are determined by a set of neural network functions. These functions are trained to maximize expected squared jump distance.
This works because each variable (of the state space) is modified in turn, so that the resulting update is invertible, with a tractable transformation inspired by Dinh et al 2016.
Overall, I believe this paper is of good quality, clearly and carefully written, and potentially accelerates mixing in a state-of-the-art MCMC method, HMC, in many practical cases. A few downsides are commented on below.
The experimental section proves the usefulness of the method on a range of relevant test cases; in addition, an application to a latent variable model is provided sec5.2. 
Fig 1a presents results in terms of numbers of gradient evaluations, but I couldn't find much in the way of computational cost of L2HMC in the paper. I can't see where the number "124x" in sec 5.1 stems from. As a user, I would be interested in the typical computational cost of both "MCMC sampler training" and MCMC sampler usage (inference?), compared to competing methods. This is admittedly hard to quantify objectively, but just an order of magnitude would be helpful for orientation. 
Would it be relevant, in sec5.1, to compare to other methods than just HMC, eg LAHMC?
I am missing an intuition for several things: eq7, the time encoding defined in Appendix C
Appendix Fig5, I cannot quite see how the caption claim is supported by the figure (just hardly for VAE, but not for HMC).
The number "124x ESS" in sec5.1 seems at odds with the number in the abstract, "50x".
# Minor errors
- sec1: "The sampler is trained to minimize a variation": should be maximize
"as well as on a the real-world"
- sec3.2 "and 1/2 v^T v the kinetic": "energy" missing
- sec4: the acronym L2HMC is not expanded anywhere in the paper
The sentence "We will denote the complete augmented...p(d)" might be moved to after "from a uniform distribution" in the same paragraph. 
In paragraph starting "We now update x":
    - specify for clarity: "the first update, which yields x' "/ "the second update, which yields x''  "
    - "only affects $x_{\bar{m}^t}$": should be $x'_{\bar{m}^t}$  (prime missing)
    - the syntax using subscript m^t is confusing to read; wouldn't it be clearer to write this as a function, eg "mask(x',m^t)"?
    - inside zeta_2 and zeta_3, do you not mean $m^t" and $\bar{m}^t$ ?
- sec5: add reference for first mention of "A NICE MC"
- Appendix A: 
    - "Let's" -> "Let"
    - eq12 should be x''=...
- Appendix C: space missing after "Section 5.1"
- Appendix D1: "In this section is presented" : sounds odd
- Appendix D3: presumably this should consist of the figure 5 ? Maybe specify.