Quantization based methods are popular for solving large scale maximum inner product search problems. However, in most traditional quantization works, the objective is to minimize the reconstruction error for datapoints to be searched. In this work, we focus directly on minimizing error in inner product approximation and derive a new class of quantization loss functions. One key aspect of the new loss functions is that we weight the error term based on the value of the inner product, giving more importance to pairs of queries and datapoints whose inner products are high. We provide theoretical grounding to the new quantization loss function, which is simple, intuitive and able to work with a variety of quantization techniques, including binary quantization and product quantization. We conduct experiments on public benchmarking datasets http://ann-benchmarks.com to demonstrate that our method using the new objective outperforms other state-of-the-art methods. We are committed to release our source code.
The paper proposes new loss functions for quantization when the task of interest is maximum inner product search (MIPS).
The paper is well written with clear descriptions, fairly comprehensive analysis and empirical exploration, and good results, and in general I agree that learning quantization so as to minimize quantization related errors on task at hand is a good strategy.  
Specific comments and suggestions for strengthening the paper are:
a) The proposed loss function in (2) includes a weight function that serves as a proxy for the task objective of giving more emphasis to quantization errors on samples with larger inner product.  Instead, why not use the true task objective which for the MIPS task is stated in the Introduction section?  If this was considered please comment on reasons for not including / discussing this in the paper, otherwise perhaps this’ll be good to discuss.
b) Did the authors consider using a task dependent training data set which will capture both ‘q’ and ‘x’ distributions and potentially lead to even further improved quantization?  This has the disadvantage of making quantization dependent on query distribution, but in cases where such data is available it will be very valuable to know if incorporating data distributions in quantization process helps performance and to what extent.
c) It will also be valuable to consider the closely related task of cosine distance based retrieval and comment on how that impacts the modifications of loss functions.
d) The idea of learning quantization under objective of interest using observed data distribution has been studied earlier (e.g. see Marcheret et al., “Optimal quantization and bit allocation for compressing large discriminative feature space transforms,” ASRU 2009), perhaps worth citing as related work.