Paper:
Pairwise Choice Markov Chains (PCMC) have been recently introduced to overcome limitations of choice models based on traditional axioms unable to express empirical observations from modern behavior economics like context effects occurring when a choice between two options is altered by adding a third alternative. The inference approach that estimates the transition rates between each possible pair of alternatives via maximum likelihood suffers when the examples of each alternative are scarce and is inappropriate when new alternatives can be observed at test time. In this work, we propose an amortized inference approach for PCMC by embedding its definition into a neural network that represents transition rates as a function of the alternatives' and individual's features. We apply our construction to the complex case of airline itinerary booking where singletons are common (due to varying prices and individual-specific itineraries), and context effects and behaviors strongly dependent on market segments are observed. Experiments show our network significantly outperforming, in terms of prediction accuracy and logarithmic loss, feature engineered standard and latent class Multinomial Logit models as well as recent machine learning approaches.

Review:
1.The goal of the paper is to connect flexible choice modeling with a modern approach to ML architecture to make said choice modeling scalable, tractable, and practical. 
2. The approach of the paper is well motivated intuitively, but could more explicitly show that PCMC-Net is needed to fix inferential problems with PCMC and that e.g. SGD and some regularization + the linear parameterization suggested by the original PCMC authors isn't scalable in itself. 
3. The approximation theorem is useful and clean, and the empirical results are intriguing. While consideration of more datasets would improve the results, the metrics and baselines considered demonstrate a considerable empirical case for this method. 
My "weak accept" decision is closer to "accept" than "weak reject." (Edit 11/25/19: I raised my score in to accept in conjunction with the author's improvements in the open discussion phase)
Improvement areas(all relatively minor):
- While I personally enjoy the choice axioms focused on by the PCMC model and this paper, stochastic transitivity, IIA, and regularity are probably more important to emphasize than Contractibility. Because the properties of UE and contractibility were not used, it may be more appropriate to use this space to introduce more of the literature on neural-nets-as-feature-embeddings stuff. 
- This paper could be improved by generalizing to a few other choice models- in particular the CDM (https://arxiv.org/abs/1902.03266) may be a good candidate for your method. This is more a suggestion for future work if you expand this promising initial result. 
- Hyper-parameter tuning: I noticed that several of your hyper parameters were set to extremal values for the ranges you considered. If you tuned the other algorithms' hyper parameters the same way, it could be the case that the relative performance is explained by the appropriateness of those ranges. Would be interesting to have a more in-depth treatment of this, but I do understand that it's a lot of work. 
Specific Notes:
Theorem 1 is nice, and the proof is clean, but doesn't explicitly note that a PCMC model jointly specifies a family of distributions \pi_S for each S \in 2^U obtained by subsetting a single rate matrix Q indexed by U. It's clear that PCMC-Net will still approximate under this definition, as \hat q_ij  approximates each q_ij because \hat q_ij doesn't depend on S. While the more explicit statement is true with the same logic in the theorem, the notational choice to have "X_i" represent the "i-th" element in S is confusing at first, as e.g. X_1 is a different feature vector for S = {2,3} and S={1,3}. I don't see this issue as disqualifying, but it took me a while to realize that there wasn't more than a notational abuse problem when I returned to the definitions where the indexing depended on the set S under consideration. 
Typos/small concerns:
-Above equation (1), the number of parameters in Q_S is |S|(|S|-1) rather than (|S|-1)^2, as each of the |S| alternatives has a transition rate to the other |S|-1 alternatives. 
-Below equation (3), I think you mean j \in S rather than 1<= j <= |S|, as S may not be {1,2,...,|S|}. Later I noticed that you always index S with {1,\dots,|S|}, but using i \in S in combination with 1<=j<=|S| was a bit confusing. 
-X_i as the i-th element of S is a bit of an abuse of notation, as it surpasses dependence on S
-In Figure 1, you show X_0 in a vector that is referred to as "S." It is my understanding that X_0 represents user features. As the user is not in the set, this is confusing. The use of a vertical ellipsis to connect \rho(X_0) to \rho(X_1) is also confusion, as \rho(X_1) is input into the Cartesian product while X_0 is input into the direct sum. 
Overall, nice job! Really enjoyed the paper and approach, good to see connections made between these literatures so that progress in discrete choice can be used at scale.