__ Summary and Contributions__: This paper studies lossy compression technique for encoders. Unlike popular previous strategy to train the encoder with a differentiable approximation of the hard quantization by adding uniform noise, this paper first proposes to use universal quantization in both training and testing phase, which avoids the train-test mismatch issue. To deal with cases where one still wants to use quantization for training, the authors propose to use a smooth approximation to quantizing function based on universal quantization scheme, and provide a method to reduce the variance of its gradient computation. Finally, the authors provide experiments on Kodak dataset showing that with the proposed strategy, we can get lower PSNR and test loss.

__ Strengths__: 1. The topic is useful in practice with many applications. It's will be interesting for many communities.
2. The ideas of adopting universal quantization in neural compression and replacing quantizing functions with a smooth surrogate are novel.

__ Weaknesses__: The motivation is not clear enough for me. In other words, it seems that the result does not resolve the problem addressed. On the upside, it is absolutely a good attempt to apply universal quantization to this compression problem. However, the main concern is the effectiveness of this approach in practice. From Figure 2 and Figure 3, we see that using solely the proposed universal quantization (UN+UQ) is much worse than prior method directly using hard quantization (UN+Q). And, the performance of using additional soft rounding (UN+UQ+SR) is just slightly better than UN+Q, especially on the complicated model where the performances are almost the same. However, SR method introduces two more parameters (\alpha for s and r) which is harder to tune, while UN+Q is simpler.
One of the major motivations of using UQ is to avoid the train-test mismatch, but from the experiments mentioned above, it performs even worse. This actually undermines the motivation.
Additionally, the experiemnts are conducted only on one dataset, which makes the results less convincing.
Hence, I would say that though applying UQ to encoder compression is new, the improvement of the paper is more or less marginal compared to previous work.

__ Correctness__: Yes.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: 1. It seems that function h is not defined consistently, e.g. eqn (7) and line 182.
2. I believe the soft rounding strategy should not be restricted to the UN and UQ scheme. Can we directly apply it to hard quantization in both training and testing, e.g. Q+SR? This is simpler and seems natural, which can also fix ''train-test mismatch''.
3. How is \alpha in s_\alpha and r_\alpha chosen, for example for the experimental results provided?
4. What is s in line 204? What does figure 1B try to say? I see that the green curve is far from the true gradient.
5. For the gradient variance reduction part, is upadting using the expectation driven by your empirical observation? Has similar idea shown in any previous work?
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After reading the rebuttal, I'm still not convinced by the improvement brought by the proposed method. Also, there surely are more datasets that can be used for testing this problem. Thus, I would keep the score as 5.

__ Summary and Contributions__: Neural network based compressors usually apply additive uniform noise during training as a proxy for the quantization that is performed during test-time. This creates a mismatch between the training and testing phases. This work proposes to instead apply universal quantization at test time thus eliminating the mismatch between training and test phases while maintaining a differentiable loss function. It is based on the fact that adding uniform noise to an input x is equivalent to subtracting a uniform random variable from x, rounding the result and then adding the same uniform random variable back. As a result, by sharing a random seed across the encoder and decoder we can easily implement universal quantization for neural network based compressors. The authors show that this is instance of a more general problem of efficiently communicating samples, which is computationally hard without distributional assumptions, but simple and practical for the uniform noise case.
While this framework bypasses the need for quantization, the authors argue that there are still scenarios where one may desire hard quantization. As direct rounding does not allow for gradient based optimisation, the authors propose to instead use a soft rounding function that has a hyperparameter alpha; small values for alpha make the soft-rounding behave like an identity and large values of alpha make the soft-rounding behave like a hard one. Instead of directly applying their soft-rounding function on an input y, which is invertible and can lead to memorisation in the decoder, the authors further add uniform noise to the soft rounded value and then perform an MSE optimal reconstruction of the original y. Finally, the authors note that the variance of the gradients of the soft rounding function can be high for large values of alpha, and they propose a way to “marginalize” the randomness for a part of the gradient expression, which empirically leads to more stable optimisation.
The authors then evaluate both universal quantization and universal quantization with soft-rounding to the Kodak dataset with a simple Linear model as well as a more flexible Mean & Scale Hyperpior model from prior literature. The results show that in general additive uniform noise with test-time hard quantization works better that universal quantization, however by incorporating soft-rounding they are able to improve, albeit slightly, upon the additive noise + test-time quantization setting.

__ Strengths__: This work has contributions in two main themes which are relevant (for a part of) the NeurIPS community; compression with and without quantization. For the first point, the authors present soft rounding which allows for gradients to flow through, along with an approximate marginalisation of the noise in the gradients that can reduce their variance. For the second point, the authors employ the concept of universal quantization, which is simple, practical and easy to implement in existing frameworks. Both of these contributions can be valuable for a broad range of research in both data as well as model compression. The application of universal quantization as well as the approximate marginalisation for the gradients are novel contributions.

__ Weaknesses__: While this paper is solid from a theoretical standpoint, I find that the empirical evaluation / experimental results are lacking. The premise of the paper seemed to imply that closing the gap between the training and test time behaviour of the algorithm, would be beneficial but unfortunately this does not seem to be the case. Additive uniform noise together with test-time hard-quantization seems to be better than universal quantization, implying that the mismatch of the train and test phases is not detrimental performance wise. Furthermore, while the addition of quantization to the universal quantization procedure seems to close the gap and improve upon the vanilla setting, the improvements seem to be small for the larger models which are typically employed, which makes me wonder whether for sufficiently large architectures universal quantization + soft rounding is necessary. Furthermore, evaluation on other datasets, such as BSD100 or Urban100, would make the experimental section of this work stronger.
Finally, I would appreciate if the authors could elaborate a bit on the bias of the approximation done at eq. 19; does this assumption hold in practice? You could for example check the errors of your expression to the one where you use eq. 18 and average over multiple samples from the the original gradient (ideally at multiple stages of training).

__ Correctness__: This work is technically correct.

__ Clarity__: The writing of the paper could use some more work, as I often found it hard to follow. Up until page 3 the text reads nicely and is clear, but after that I noticed a couple of things:
- The paragraph from line 141 to 146 is unclear and I would appreciate if the authors could explain it a bit better. What do you mean with the “single coefficient which is always zero”? Does such a case happen in practice?
- At the intro of section 4 you mention that you will show that quantization is a limiting case of universal quantization if you allow for flexible encoders and decoders, but I don’t see any discussion on this part in the rest of the section. You rather introduce and work on round(round(y) + u) which is different from the round(y - u) + u of universal quantization.
- What do you mean with the statement, “the prior is smooth enough to be approximately uniform in each interval“ at line 168-169? You mean that it is flat inside that interval?
- At line 182 you mention that h is a differentiable function but later at line 187-188 you mention that you can evaluate the derivative of the expectation even when h is not differentiable (contradicting the previous assumption).
Furthermore, I believe that more details about how the uniform noise + universal quantization + soft rounding setting works are appropriate:
- Is the soft rounding at training time reverted to a hard-rounding at test time?
- Where exactly is the universal quantization applied? At the intermediate result of soft_round(y) + u?
- What happens when you employ soft rounding without universal quantization, i.e. soft_round(y) + u for training and just round(y) for testing?

__ Relation to Prior Work__: The relation to prior work is generally good. Perhaps some interesting points for the authors is that the proposed rounding function seems to be similar / same as the one at [1] and that soft rounding of a noisy input has been also explored at [2].
[1] Differentiable Soft Quantization: Bridging Full-Precision and Low-Bit Neural Networks, Gong et al.
[2] Relaxed Quantization for Discretized Neural Networks, Louizos et al.

__ Reproducibility__: No

__ Additional Feedback__: Typos and nitpicks:
- Line 59-60, the correspondence to VAE is true for any distortion metric that corresponds to a log probability and not just the mean-squared error.
- Line 116, “statistically efficiently” -> “statistically efficient“
- Line 137, “density that is” -> “density that”
- Line 240, “Training”
After reading the authors rebuttal
I appreciate the authors summarisation of the contributions and clarifications; after reading them, I decided to keep my score the same. The work is definitely interesting and worthwhile to be considered for acceptance. The things that I would like to see in order to fully recommend acceptance would be the following:
1. You mention that universal quantization has the potential for big future improvements on other metrics, beyond PSNR, and mention a couple of other choices. While this could end up being the case, it could also end up being similar to the conclusions from the train / test mismatch we saw in this work. As a result, I would appreciate experimental evidence to satisfy these claims.
2. Exploration of the bias in the gradients introduced by performing the approximation at eq. 19.
3. Rephrasing the abstract and pitch of paper, as currently it gives the impression that the train / test gap is a problem, when in practice it is not.
4. _x0010_Improving the clarity of the paper, e.g, at the intro of section 4 you mention that you will show that quantization is a limiting case of universal quantization if you allow for flexible encoders and decoders, but I don’t see any discussion on this part in the rest of the section.

__ Summary and Contributions__: Authors describe two methods for handling quantization (a discontinuous operation) in learned systems, by 1) implementing a form of Ziv's Universal Quantization for transmitting noisy continuous values, and 2) introducing smoothed (and thus differentiable) quantizers. They examine performance of the first, and the combination of the two, on a simple linear coder, and on one recently published (Minnen et al, 2018).

__ Strengths__: Focused and well-written, the paper builds on recent advances in deep CNN compression, in particular Balle et al 2017 and Minnen etal 2018. Those papers introduced a uniform noise approximation to the quantizer, so as to allow for a continuous and differentiable rate+distortion loss function during training, but then reverted to quantization for testing. Here the authors make use of Ziv's result to provide a direct encoding of the uniform noise values, thus allowing the test phase to be fully consistent with the training. i think this is a nice contribution (and was happy to learn about Ziv's result, which I'd not seen). The use, in addition, of a "softened" differentiable quantizer (as proposed in Agustsson 2019) leads to better performance.

__ Weaknesses__: * I was surprised to see, after the authors touted advantages of using a consistent training and test implementations, that the results of the (UN+UQ) system were significantly worse than those of the "inconsistent" solution introduced by Balle et al 2017 (UN+Q). Only when the softened quantizer is added (UN+UQ+SR) do we see a relatively small improvement. Why? Are there potential ways to improve this?
* This makes one wonder about how much the UQ noise matters. In particular, it would be instructive to see a comparison to (UN+SR). Given the previous comment, one might suspect this would lead to even better performance - and thus that the UQ methodology, despite its mathematical interest, is not of practical value.
* Although I think it's important and interesting, this is a pretty heavy and narrowly-focused topic for the NeurIPS community.

__ Correctness__: Math generally solid.

__ Clarity__: Generally clearly written, especially for such a specialized technical topic.

__ Relation to Prior Work__: Solid bibliography.

__ Reproducibility__: Yes

__ Additional Feedback__: >>> Added after reading author's feedback, and other reviewer comments:
I appreciate the conceptual and mathematical points of the paper, and think they are valuable for the compression-interested sub-community at NeurIPS. I also appreciate the value of quantifying the train/test mismatch that arises from the uniform noise approximation (even thought it comes out relatively small). On the other hand, I see no evidence that "universal quantization has the potential to lead to much bigger improvements in the future." And consistent with R2, I think the authors should re-frame the initial statement (in abstract/intro) of the contributions. Overall, I did not find the feedback or other reviews significantly altered my view of the paper, which is above threshold. In any case, I hope the authors will find our comments helpful in improving the work.

__ Summary and Contributions__: This paper proposes an approach to account for quantization noise based on Ziv's universal quantization principles.

__ Strengths__: The addressed problem is relevant and timely and may have potentially a broad impact.
The paper is very clear and very well written. The mathematical explaination is also clear, even though it owes a lot to Ziv.

__ Weaknesses__: Experiments with models less complex than scalable hyperpriors and more complex than linear would have given a better understanding of the effectiveness of the technique. For example, the authors could experiment with a simple autoencoder with a residual encoder.
It is not totally clear how to practically apply the propsoed scheme due to the lack of a section that explains how to apply it, which may limits the reprodicibility of teh results.
This work is limited by definition to uniform noise.
What is the extra training complexity introduced by the proposed method? can it be quantified ?
The experiments are performed on one dataset only (altough large), which is limiting.
Is the source code provided?
There are some typos, like for example in the title of sec. 5.2 "Traininig"

__ Correctness__: Yes, albeit stronger experimental evaluation is welcome.

__ Clarity__: Very well written.

__ Relation to Prior Work__: Prior work is discussed appropriately.

__ Reproducibility__: No

__ Additional Feedback__: