text: In this work, discrete-weight NNs are trained using the variational Bayesian framework, achieving similar results to other state-of-the-art models. Weights use 3 bits on the first layer and are ternary on the remaining layers. - Pros: The paper is well-written and connections with the literature properly established. The approach to training discrete-weights NNs, which is variational inference, is more principled than previous works (but see below). - Cons: The authors depart from the original motivation when the central limit theorem is invoked. Once we approximate the activations with Gaussians, do we have any guarantee that the new approximate lower bound is actually a lower bound? This is not discussed. If it is not a lower bound, what is the rationale behind maximizing it? This seems to place this work very close to previous works, and not in the "more principled" regime the authors claim to seek. The likelihood weighting seems hacky. The authors claim "there are usually many more NN weights than there are data samples". If that is the case, then it seems that the prior dominating is indeed the desired outcome. A different, more flat prior (or parameter sharing), can be used, but the described reweighting seems to be actually breaking a good property of Bayesian inference, which is defecting to the prior when evidence is lacking. In terms of performance (Table 1), the proposed method seems to be on par with existing ones. It is unclear then what the advantage of this proposal is. Sparsity figures are provided for the current approach, but those are not contrasted with existing approaches. Speedup is claimed with respect to an NN with real weights, but not with respect existing NNs with binary weights, which is the appropriate baseline. - Minor comments: Page 3: Subscript t and variable t is used for the targets, but I can't find where it is defined. Only the names of the datasets used in the experiments are given, but they are not described, or even better, shown in pictures (maybe in a supplementary). The title of the paper says "discrete-valued NNs". The weights are discrete, but the activations and outputs are continuous, so I find it confusing. As a contrast, I would be less surprised to hear a sigmoid belief network called a "discrete-valued NN", even though its weights are continuous.