REx: Data-Free Residual Quantization Error Expansion

Deep neural networks (DNNs) are ubiquitous in computer vision and natural language processing, but suffer from high inference cost. This problem can be addressed by quantization, which consists in converting floating point operations into a lower bit-width format. With the growing concerns on privacy rights, we focus our efforts on data-free methods. However, such techniques suffer from their lack of adaptability to the target devices, as a hardware typically only support specific bit widths. Thus, to adapt to a variety of devices, a quantization method shall be flexible enough to find good accuracy v.s. speed trade-offs for every bit width and target device. To achieve this, we propose REx, a quantization method that leverages residual error expansion, along with group sparsity and an ensemble approximation for better parallelization. REx is backed off by strong theoretical guarantees and achieves superior performance on every benchmarked application (from vision to NLP tasks), architecture (ConvNets, transformers) and bit-width (from int8 to ternary quantization).

Paper

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Peer review

Reviewer m1LK5/10 · confidence 3/52023-06-30

Summary

The paper proposed REx which allows the flexibility to find the PTQ recipe given a speed/accuracy tradeoff by computing the residual expansion of weights and activations in a data-free manner. The method minimizes error between the FP16 and INT quantizations by computing and adding the quantized residual errors. The authors further propose to reduce the overhead with selective error computations by using the parameter magnitude as a proxy for parameter importance. Additionally, the authors also provide the theoretical error upper bound for a given expansion and show the tightness of the bound. ### Post Rebuttal Update I have increased my score to 5 -- Borderline accept.

Strengths

The proposed PTQ method is data-free and the idea is quite interesting. The authors provide theoretical backing for the error upper bound for a given expansion which is greatly appreciated The paper compares with a fair number of previous approaches

Weaknesses

Figure 2 is extremely difficult to read due to the color scheme (especially for someone who might be colorblind). I would sincerely request to authors to improve the color scheme. The authors claim that the residual expansions can be made sparse using the norm of the parameters as a selection criterion which might lead to unstructured sparsity. Especially in Table 1, where the sparsity is 50%, this would lead to high inference latencies unless the target device supports unstructured sparsities (which current GPUs do not). Why would this method be preferred over existing methods? I am not sure I understand the BOPS metric. Hardware targets only support certain bit widths which have a constant cost of operations i.e INT4 cores would take the same amount of time even if you use W3/A3 recipe. I don't think comparing existing methods at equivalent BOPS is fair. For LLMs, the authors claim that the residue with W1/A16 has virtually no cost. I am not sure I understand I follow this, even with 1bit residue -- Assuming someone has optimized the weight loading with cache-line optimizations that have minimal impact on weight loading times, performing a large number of FP16 operations will add significant computational overhead when the models are compute-bound. In Line 54, the authors claim a budget of $\gamma$ for the computational overhead but instead use a budget of $\gamma$ for the weight overhead. I would like to point out that a given weight over-head budget does not translate to an equal computational overhead [1], which is usually larger. In general, there is no discussion on the impact of inference latencies in using this strategy which *significantly* diminishes the impact of this work. The writing of the paper can be improved. Some of the errors I noticed were: shall -> should in line 8, in -> of in line 83 [1] - https://dl.acm.org/doi/abs/10.1145/3400302.3415679

Questions

I am not convinced about BOPS and how comparing existing methods at equivalent BOPS is fair. I would like to see some results on latency using this strategy. Latency (and size) of models are the true real-world metrics and I think BOPS is a poor proxy for Latency and limits the applicability of this method. I would like to understand how the authors propose to deal with unstructured sparsity that the method introduces.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

2 fair

Limitations

NA

Reviewer mdRP5/10 · confidence 4/52023-07-04

Summary

In this paper, the authors propose a fixable quantization method called REx, which utilizes residual error expansion to further improve quantization error. Additionally, they suggest applying group-sparsity to reduce the computational cost associated with residual error expansion. Existing DFQ methods have a limitation in terms of flexibility when it comes to quantizing based on the representation format supported by hardware, considering the trade-off between accuracy and speed-up. However, the REx method overcomes this limitation and provides a better trade-off. Furthermore, by calculating the theoretical upper bound of quantization error during the process of residual error expansion and demonstrating it through actual quantization error, the REx method shows that it can have less quantization error than or similar to existing methods at fewer BOPs. The proposed REx method can be combined with recently developed quantization methods, leading to quantized models with improved accuracy.

Strengths

* It demonstrates that the proposed REx method can find a better trade-off point between compression ratio and accuracy compared to existing methods through residual error expansion and group-sparsity expansion. * Theoretical and experimental evidence is provided for the upper bound of quantization error that the proposed REx method can achieve, showing a theoretical basis for achieving better quantization error at the same BOPs. * The proposed method can be applied in a composite manner to various quantization methods previously proposed, and experiments on different models demonstrate improvements in accuracy.

Weaknesses

* It appears necessary to compare the results in Table 3 with other papers on LLM quantization, such as OPTQ, LLM.int8, and SmoothQuant. * Throughout the paper, different bit-widths for weights and activations were used, which may present challenges in utilizing the formats supported by hardware. However, there is a lack of discussion regarding this aspect. * There is a need for discussion on the criteria for dividing clusters and the benefits of higher sparsity in achieving better accuracy. * In order to discuss the trade-off between accuracy and speed, there should be a discussion on speed. The paper lacks discussion on this aspect. It would be beneficial to provide unit test results comparing the latency when applying group-sparsity expansion along with INT GEMM + SpMM compared to conventional INT GEMM, as well as the throughput on the GPU.

Questions

1. While Figure 2 represents the inference time in terms of BOPs, is there any analysis regarding the actual latency? I am curious about the analysis of the additional computational cost introduced by residual expansion and how much it can be reduced by group-sparsity expansion. 2. In Figure 2 and Figure 3, when W2A8 is not supported by the hardware, additional packing and unpacking operations are required, which may incur overhead. It would be interesting to see how the graph's trends change when considering this overhead. 3. In Table 5, there is a tendency for higher group sparsity to result in better performance. I am curious about the reasons behind this observation.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

The paper provided valuable insights by demonstrating the ability of the proposed method to find a better trade-off between compression ratio and accuracy. However, it is challenging to agree with the paper's emphasis on whether it truly offers a superior approach in terms of the accuracy vs. speed trade-off. Since INT GEMM relies on the formats supported by hardware, it is necessary to analyze the speed gain achievable when applying REx + group-sparsity in practice.

Reviewer d6mY6/10 · confidence 3/52023-07-05

Summary

In this work, authors proposed a novel quantization method called REx, that leverages residual error expansion. The idea is to repeatedly apply quantization to the quantized residual error, which provides an increasingly better approximation of the original floating-point tensor as the number of steps grows. To combat the computational overhead from additional quantization steps, authors combine the residual expansion with the group sparsity. Two hyper-parameters of the method (a number of steps K, and the sparsity rate $\gamma$) are used to control the accuracy vs. speed trade-off. The method is agnostic to the quantization operator itself and can be combined with many existing methods.

Strengths

* Experiments on both CNNs (including some of the more difficult-to-quantize models like MobileNets and EfficientNets) and LLMs across various tasks + combination with several quantization operators from the prior work. * Theoretical derivations on the upper bound of the quantization error + comparison with the empirical values. * Some attention to efficient hardware implementation (Section 4.1), e.g. instead of using K separate CUDA kernels for the error expansion step, use a single one with concatenated output channels, etc.

Weaknesses

* Two hyper-parameters (K and $\gamma$) that have to be set. Even though $\gamma$ can be set to match a certain BOP target (as it is done in the paper), the user still has to select a value for K. * I have some concerns related to the data-free nature of the method, please see and elaborate on the below question. * I would like to see error bars / standard deviations in some of the results, e.g. in Table 2. I wonder if the improvement is statistically significant (it is well known that some of the datasets from GLUE are quite small which might lead to high standard deviations in the results).

Questions

* The method is advertised as data-free, however, I don't see how it is the case in general. As far as I understood, it's data-free if the quantization operator Q itself is data-free (which is the case of DFQ, for instance). However, in most experiments, the underlying Q is not data-free (e.g., AdaRound, BRECQ, or even naive quantization with static range estimation). Could you elaborate? * Related to the previous comment, in the group sparsity, the importance of a neuron is defined by the norm of its weights (L151: assuming the data-free setting). Assuming we have access to activations and gradients, have you experimented with other importance metrics (e.g., gradient-based, FIT)? * I Would like to know a bit more details on the outlier quantization in LLMs. How are outliers detected/defined? Will the suggested approach work with outliers that are both positive and negative? In Table 3, what gives the most improvement - the method itself or the special treat of the outliers (would be nice to have an ablation study and/or have some additional data on e.g. quantization error for tensors with outliers with W1 vs. W4). * (Sorry if I missed it) Do you quantize embeddings, LayerNorm weights/biases, and LLM head, are there any specific assumptions (e.g., first and or last layer in 8-bit)?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

Authors mentioned one limitation that the method does not adapt to the per-layer importance and runtime cost discrepancies.

Reviewer E57B5/10 · confidence 4/52023-07-06

Summary

This paper presents a low-bitrate post-hoc quantization method with three major techniques: residual expansion, input expansion, and sparse expansion. The techniques are theoretically guaranteed to improve the accuracy of the quantization, and experimental results show that they outperform competing approaches.

Strengths

* Provides theoretical bound for accuracy-bitrate trade-off * Evaluation that compares the theoretical bound and empirical benchmarks.

Weaknesses

Lack of (machine) runtime evaluation. The project propose to use quantization for better inference runtime, yet none of the benchmark demonstrates the improvement.

Questions

What is the (machine) runtime on CPU and/or GPU?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

2 fair

Presentation

3 good

Contribution

3 good

Limitations

This paper uses public dataset and benchmark, so it would not have potential negative societal impact.

Reviewer E57B2023-08-12

It would be helpful if the authors could expand the runtime evaluations to match the benchmark with bops comparisons. The authors indicate that the bops metric is a good proxy for real-world performance, but this connection is not well-supported. Even if the bops metric does provide a good proxy for actual performance, the precise definition of bops is lacking in this work. For future researchers to reference, please provide the formulation of bops.

Authorsrebuttal2023-08-12

Response to reviewer E57B

We thank the reviewer for their comment and consideration of the provided runtimes. To address your questions. We use the generic definition of the bitwise operations as in [1,2,3]. This is the standard in quantization and we will include these references to the manuscript for future researcher. Regarding the connection between runtime and bops. The goal of the shared results is to highlight that in the context of REx and other standard quantization approaches such as DFQ. The results that were given for equivalent bops (Table 1 in the paper) do translate in very close runtime performance as shown in the common rebuttal. Similarly our results for LLMs (table 3 in the paper) at equivalent bops also translate in almost identical runtime as shown in the second table of the rebuttal. We hope that this message clarifies our previous responses. In short, our rebuttal shows that, in the context of the benchmarks conducted in our study at equivalent bops, the empirical latencies of the proposed REx is indeed almost identical (or even lower) to other quantization schemes while achieving a higher accuracy. ## References [1] Van Baalen, Mart, et al. "Bayesian bits: Unifying quantization and pruning." Advances in neural information processing systems 33 (2020): 5741-5752. [2] Cai, Zhaowei, and Nuno Vasconcelos. "Rethinking differentiable search for mixed-precision neural networks." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2020. [3] Nikolić, Miloš, et al. "Bitpruning: Learning bitlengths for aggressive and accurate quantization." arXiv preprint arXiv:2002.03090 (2020).

Reviewer E57B2023-08-20

In [1], the authors only count the number of multiplication operations. However, in [3] all bit manipulations are equally attributed. > …the BOP count measures the number of multiplication operations multiplied by the bit widths of the operands. To compute the BOP count we use the formula introduced by [3], but ignore the terms corresponding to addition in the accumulator since its bit width is commonly fixed regardless of operand bit width. Several natural questions arise, such as how to derive the number of BOPs for lookup tables (LUTs) that are commonly used in quantization. The BOPs count for sparse memory loading is also undefined in all these references. Memory operations are much slower than regular bitwise operations on modern architectures, yet there was no discussion on how this is addressed with BOPs. Inconsistent BOPs definitions are a major concern, and the exact formulation was not listed in the supplementary materials. Due to these considerations, I would like to keep my original ratings.

Authorsrebuttal2023-08-21

response to reviewer E57B

We understand the concern regarding the definition of the BOP count and will add an exact definition to the supplementary material as suggested shall the paper be accepted. Regarding the few points mentioned: 1. on modern hardware, the bit-width of the accumulation is fixed and we do account for it in our measurements due to the reduction operation from REx (this is thoroughly benchmarked in appendix E). 2. the number of BOPs for look-up table is a challenge for non-uniform quantization as we do not of any uniform quantization method that requires LUTs. 3. BOPs with sparsity: we do agree with the fact that this metric is not well suited for sparsity. However, in REx we do not use BOPs to evaluate the unstructured spare expansion which is applied only to LLMs. In that case, we measure the memory foot-print which is the most important challenge to LLM inference and reported latency measurements in our rebuttal which show that empirically, REx achieves state-of-the-art performance by a significant margin in terms of accuracy at the cost of 0.26% of the latency. Overall, the goal of these metrics is to showcase the low cost of REx which offers new trade-offs in terms of accuracy versus speed which is well supported by the combination of our results from the rebuttal with the performance originally listed in the paper.

Reviewer m1LK2023-08-14

I thank the authors for the clarification. I read through all the other reviews and the authors' replies -- the comparison with Smoothquant is also appreciated. The latency numbers look good and are much more convincing than BOPS. > For LLMs, the authors claim that the residue with W1/A16 has virtually no cost. I am not sure I understand I follow this, even with 1bit residue -- Assuming someone has optimized the weight loading with cache-line optimizations that have minimal impact on weight loading times, performing a large number of FP16 operations will add significant computational overhead when the models are compute-bound. This still looks misleading and should be modified based on the response posted by the authors' describing the specific circumstances in which it is true. > In Line 54, the authors claim a budget of $\gamma$ for the computational overhead but instead use a budget of $\gamma$ for the weight overhead. I would like to point out that a given weight over-head budget does not translate to an equal computational overhead [1], which is usually larger. Can the authors please clarify this? I would also like the authors to mention the speedup for unstructured sparse expansion is dependent on the target hardware. Based on all the additional evidence, I would like to increase the score to 5.

Authorsrebuttal2023-08-14

Response to Reviewer m1LK

We would like to thank the reviewer for their constructive feedback and reactivity. We agree with all the remarks that were made and will make the requested changes, shall the paper be accepted for publication. In the specific context of REx, the budget $\gamma$ for weight overhead is very similar to the corresponding computational overhead, due to: 1. the proposed method increases the layer widths without increasing the output tensors size (due to the reduction from summing the residuals after each layer). 2. the extra computations have an appropriate structure, e.g. a 50\% overhead from REx on a layer with a power of $2$ neurons, will add a number of computations that is often well supported by the device. Empirically, we have measured the induced latency from a small budget $\gamma < 75$% (at least $25$% sparsity) and seen that the parallelization capacities of modern hardware lead to a smaller latency overhead. We will also add this element to the implementation section of the revised manuscript.

Reviewer mdRP2023-08-16

Thank you for the detailed answers and results. I have read the authors' rebuttal as well as other reviews. I would like to keep my rating.

Reviewer d6mY2023-08-21

Thanks a lot for clarifying my questions and commenting on my concerns. I do appreciate to see that the results are significant and also seeing the actual speed improvement from the general response. I read over all other reviews and the corresponding responses. Overall, considering all strength and weaknesses pointed out by various reviewers, I believe the strength do outweigh some of the weaknesses and therefore still lean towards accept. Therefore I keep my rating of weak accept.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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