Hybrid and Non-Uniform DNN quantization methods using Retro Synthesis data for efficient inference
Authors: TEJPRATAP GVSL
Raja Kumar
Pradeep NS
Venue:  (2021)
Keywords: quantization
dnn inference
data free quantization
synthetic data
model compression

Abstract: Existing post-training quantization methods attempt to compensate for the quantization loss by determining the quantized weights and activation ranges with the help of training data. Quantization aware training methods, on the other hand, achieve accuracy near to FP32 models by training the quantized model which consume more time. Both these methods are not effective for privacy constraint applications as they are tightly coupled with training data. In contrast, this paper proposes a data-independent post-training quantization scheme that eliminates the need for training data. This is achieved by generating a faux dataset hereafter called as $\textit{‘Retro-Synthesis Data’}$ from the FP32 model layer statistics and further using it for quantization. This approach outperformed state-of-the-art methods including, but not limited to, ZeroQ and DFQ on models with and without batch-normalization layers for 8, 6 and 4 bit precisions. We also introduced two futuristic variants of post-training quantization methods namely $\textit{‘Hybrid-Quantization’}$ and $\textit{‘Non-Uniform Quantization’}$. The Hybrid-Quantization scheme determines the sensitivity of each layer for per-tensor and per-channel quantization, and thereby generates hybrid quantized models that are $10 - 20\%$ efficient in inference time while achieving same or better accuracy as compared to per-channel quantization. Also this method outperformed FP32 accuracy when applied for models such as ResNet-18, and ResNet-50 onImageNet dataset. In the proposed Non-Uniform quantization scheme, the weights are grouped into different clusters and these clusters are assigned with a varied number of quantization steps depending on the number of weights and their ranges in respective cluster. This method resulted in an accuracy improvement of $1\%$ against state-of-the-art quantization methods on ImageNet dataset.

Under review as a conference paper at ICLR 2021
HYBRID AND NON-UNIFORM, QUANTIZATION METH- ODS USING RETRO SYNTHESIS DATA FOR EFFICIENT IN- FERENCE
Anonymous authors Paper under double-blind review
ABSTRACT
Existing quantization aware training methods attempt to compensate for the quan- tization loss by leveraging on training data, like most of the post-training quan- tization methods, and are also time consuming. Both these methods are not ef- fective for privacy constraint applications as they are tightly coupled with training data. In contrast, this paper proposes data inta-independent post-training quantiza- tion scheme that eliminates the need for training data. This is achieved by gen- erating faux dataset, hereafter referred FP32 model layer statistics and further using it for quantization. This approach outperformed state-of-the-art methods including, but limited to, ZeroQ and DFQ on models with and without Batch-Normalizatiee
'Retr-Syythhsis Data', from the
layers for 8, 6, and 4 bit precisions on ImageNet and CIFAR- datasets. We also introduced two futuristic variants of post-tran post-training quantization methods namely 'Hybrid Quantization' and 'Non- Uniform Quantization . The Hybrid Quantization scheme determines the sensitivity of each layer for per-tensor & per-channel quantization, and thereby generates hybrid quantized models that are '10 to 20% efficient inference time while achieving the same or better accuracy compared per-channel quantization. Also, this method outperformed FP32 accuracy when applied for ResNet-18. and ResNet-50 models the ImageNet dataset. In the proposed Non-Uniform Quan- tization scheme, the weights are grouped into different clusters and these clusters are assigned with varied number of quantization steps depending on the number of weights and their ranges in the respective cluster. This method resulted in '1 % accuracy improvement against state-of-the-ar methods the ImageNet dataset.
INTRODUCTION
Quantization is a widely used and necessary approach to convert heavy Deep Neural Network (DNN) models in Floating Point (FP32) format to light-weight lower precision format, compatible with edge device inference. The introduction of lower precision computing hardware like Qualcomm
Hexagon DSP (Codrescu 2015) resulted in various quantization methods (Morgan al. 1991. Rastegari et al. 2016 Wu et al. 2016 Zhou et al.] 2017] Li et al.) 2019 Dong et al.] 2019 Krish- namoorthi 2018) compatible for edge devices. Quantizing FP32 DNN INT8 lower precision results in model size reduction by at least 4XX based on the precision opted for. Also, since the com- putations happen in lower precision, it implicitly results in faster inference time and lesser power consumption. The above benefits with quantization come with caveat of accuracy loss, due to noise introduced in the model's weights and activations.
In order to reduce this accuracy loss, quantization aware fine-tuning methods are introduced (Zhu al., 2016) /Zhan et al., 2018 Choukroun et al. 2019 Jacob et al., 2018), [Baskin al., 2019 Courbariaux et al. 2015). wherein the FP32 model is trained along with quantizers and quantized weights. The major disadvantages of these methods are, they are computationally intensive and time-consuming since they involve the whole training process. To address this, various post-training quantization methods (Morgan et al. 1991 Wu et al. 2016 Li et al. 2019 Banner et al. 2019) are developed that resulted in trivial to heavy accuracy loss when evaluated
different DNNs. Also, to determine the quantized model's weight and activation ranges most of these methods require access to training data, which may not be always available in case of applications with security and privacy

---

Under review as a conference paper at ICLR 2021
constraints which involve card details, health records, and personal images. Contemporary research
in post-training quantization (Nagel et al., 2019; Cai et al. 2020) eliminated the need for training data for quantization by estimating the quantization parameters from the Batch-Normalization (BN) layer statistics of the FP32 model but fail to produce better accuracy when BN layers are not present
in the model.
To address the above mentioned shortcomings, this paper proposes a data-independent post-training quantization method that estimates the quantization ranges by leveraging on 'retro-synthesis data generated from the original FP32 model. This method resulted in better accuracy as com- pared to both intepedndden and data-dependent state-of faate-of-heearr art quantization methods on models ResNet18, 8, ResNet50 (He et al., (2016), MobileNetV2 (Sandler et al.] 2018)) AlexNet (Krizhessky) et al. 2012) and ISONet (Qi et al. 2020) on ImageNet dataset (Deng et al. 2009). It also out- performed state-of-the-art methods even for lower precision such as 6 and 4 bit on ImageNet and CIFAR-10 datasets. The 'retro-synthesis' data generation takes only 10 to /2 sec of time to gener- ate the entire dataset which is minimal overhead as compared to the benefit of data independence it provides. Additionally, this paper introduces two variants of post-training quantization methods namely 'Hybrid Quantization' and 'Non- Moo--unformm Quantization'.
2
PRIOR ART
2.1 QUANTIZATION AWARE TRAINING BASED METHODS
An efficient integer only arithmetic inference method for commonly available integer only hardware is proposed in Jacob et al. (2018), wherein a training procedure is employed which preserves the accuracy of the model even after quantization. The work in Zhang et al. (2018) trained a quantized bit compatible DNN and associated quantizers for both weights and activations instead of relying on handcrafted quantization schemes for better accuracy. A "Trained Ternary Quantization' approach is proposed in Zhu et al. (2016) wherein the model is trained to be capable of reducing the weights to 2-bit precision which achieved model size reduction by 16x without much accuracy loss. Inspired by other methods [askinn al. (2019) proposes a 'Compression Aware Training' scheme that trains a model to learn compression of feature maps in a better possible way during inference. Similarly, in binary connect method (Courbariaux et al. 2015) the network is trained with binary weights during forward and backward passes that act as regularizer. Since these methods majorly adopt training the networks with quantized weights and quantizers, the downside of these methods is not only that they are time-consuming but also they demand training data which is not always accessible.
2.2 POST TRAINING QUANTIZATION BASED METHODS
Several post-training quantization methods are proposed to replace time-consuming quantization aware training based methods. The method in Choukroun et al. (2199) avoids full network training, by formalizing the linear quantization as 'Minimum Mean Squared Error' and achieves better accu- racy without retraining the model. 'ACIQ' method (Banner et al. 2019) achieved accuracy close to FP32 models by estimating analytical clipping range of activations in the DNN. However, to com- pensate for the accuracy loss, this method relies on a run-time per-channel quantization scheme for activations which is ineffieient and not hardware friendly. In similar lines, the OCS method (Zhao) et al. 2019) proposes to eliminate the outliers for better accuracy with minimal overhead. Though these methods considerably reduce the time taken for quantization, they are unfortunately tightly coupled with training data for quantization. Hence they are not suitable for applications wherein ac- cess to training data is restricted. The contemporary research on data free post-training quantization methods was successful in eliminating the need for accessing training data. By adopting a per-tensor quantization approach, the DFQ method (Nagel et al., 2019) achieved accuracy similar to the per- channel quantization approach through cross layer equalization and bias correction. It successfully eliminated the huge weight range variations across the channels in a layer by scaling the weights for cross channels. In contrast ZeroQ (Cai et al. 2020) proposed quantization method that eliminated the need for training data, by generating distilled data with the help of the Batch-Normalizate layer statistics of the FP32 model and using the same for determining the activation ranges for quan- tization and achieved state-of-th the--rrrrrt accuracy. However, these methods tend to observe accuracy degradation when there no narta--oomalizattion layers present in the FP32 model.
2

---

Under review as a conference paper at ICLR 2021
To address the above shortcomings the main contributions in this paper follows:
A data-independent post-training quantization method by generating the 'Retro Synthesis' data, for estimating the activation ranges for quantization, without depending on the Batch- Normalization layer statistics of the FP32 model.
Introduced 'Hybrid Quantization' method, combination of Tensorror and Per-Chare Per-Channel schemes, that achieves state-of-the-ar accuracy with lesser inference time as compared to fully per-channel quantization schemes.
Recommended a 'Non-Uniform Quantization' method, wherein the weights in each layer are clustered and then allocated with a varied number of bins to each cluster, that achieved '1 %% better accuracy against sate-of-the-as methods ImageNet dataset.
METHODOLOGY
This section discusses the proposed data-independent post-training quantization methods namely (a) Quantization using retro-synthesis data, (b) Hybrid Quantization, and (c) Non-Uniform Quantiza- tion.
3.1 QUANTIZATION USING RETRO SYNTHESIS DATA
In general, post-training quantization schemes mainly consist of two parts (i) quantizing the weights that are static in given trained FP32 model and (ii) determining the activation ranges for layers like ReLU, Tanh, Sigmoid that vary dynamically for different input data. In this paper, asymmetric uniform quantization is used for weights whereas the proposed 'retro-synthesis' data is used determine the activation ranges. It should be noted that we purposefully chose resort to simple asymmetric uniform quantization to quantize the weights and also have not employed any advanced techniques such as outlier elimination of weight clipping for the reduction of quantization loss. This is in the interest of demonstrating the effectiveness of 'retro-synthesi data in accurately determining the quantization ranges of activation outputs. However, in the other two proposed methods (b), and (c) we propose two newly developed weight quantization methods respectively for efficient inference with improved accuracy.
3.1.1 RETRO-SYNTHHE DATA GENERATION
Aiming for -eaeenndeeendnnt quantization method, challenging to estimate activation ranges without having access to the training data. An alternative is to use "random data" having Gaussian distribution with 'zero mean' and 'unit variance' which results in inaccurate estimation of activation ranges thereby resulting in poor accuracy. The accuracy degrades rapidly when quantized for lower precisions such as 6, 4, and 2 bit. Recently ZeroQ (Cai et al. 2020) proposed quantization method using distilled data and showed significant improvement, with no results are showcasing the gener- ation of distilled data for the models without accuracy results.
layers and their corresponding
In contrast, inspired by ZeroQ (Cai et al. 2020) we put forward a modified version of the data generation approach by relying on the fact that. DNNs which are trained discriminate between different image classes embeds relevant information about the images. Hence, by considering the class loss for particular image class and traversing through the FP32 model backward, is possible to generate image data with similar statistics of the respective class. Therefore, the proposed "reiro- synthesis" " data generation is based on the property of the trained DNN model. where the image data that maximizes the class score is generated, by incorporating the notion of the class features captured by the model. Like this, we generate set of images corresponding to each class using which the model is trained. Since the data is generated from the original model itself we named the data as "retro-synthesis" data. should be observed that this method has no dependence on the presence of Batch- Noich-Normalization layers in the FP32 model, thus overcoming the downside of ZeroQ. is also evaluated that, for the models with Batch-Norma Natth-Normallzationn layers, incorporating the proposed "class-loss" functionality to the distilled data generation algorithm as in ZeroQ results in improved accuracy. The proposed "retro-synthesis" data generation method detailed in Algorithm| [ Given, fully trained FP32 model and class of interest, aim is empirically generate image that
3

---

Under review as a conference paper at ICLR 2021
representative of the class in terms of the model class score. More formally, let P(C) be the soft-max of the class C, computed by the final layer of the model for an image Thus, the aim is, to generate image such that, this image when passed to the model will give the highest softmax value for class C.
Algorithm Retro synthesis data generation
Input: Pre-determined FP32 model (M), Target class (C). Output: set retro-synthesis data corresponding to Target class (C).
1. Init: I
wawpoommaaauusiaaaaahhhhh..a input shape)
2. Init: Targel rNand(No.. of classes) ô (rgmax(Targel)) -C
3. Init: 140 0, =1
4. Get (Hs 0z) from batch norm layers of M (if present), i E 0, 0,1...., where - No.of batch norm layers
5. for 0, 1....................... No. of Epochs
(a) Forward propagate and gather intermediate activation statistics
(b) Oulpul (T)
(c) LOSSBN=0
(d) for ls = 0.1,...,
i. Get (H σk)
ii. LOSSEN t LossEN+L((Hee,O ,
(c) Calculate (1gy o') of
(f) Lossa &- L[(Ho, (o), (())
(g) Losse <- L(Target,, Output)
(h) Total loss LOSSBN + Lossg + Losse
(1) Update I t backward(Total loss)
The "retro-synthesi"" data generation for a target class C starts with random data of Gaussian dis- tribution I and performing forward pass on / to obtain intermediate activations and output labels. Then we calculate the aggregated loss that occurs between, stored batch norm statistics and the in- termediate activation statistics (LBN), the Gaussian loss (Lc), and the class loss (Lc) between the output of the forward pass and our target output. The L2 loss formulation as in equation [ is used for LaN and La calculation whereas mean squared error is used to compute Le- The calculated loss is then backpropagated till convergence thus generating batch of retro-synthesis data for a class C. The same algorithm is extendable to generate the retro-synthesis data for all classes as well.
BN L((ps, ax), (1 BN (NN) || 14 ook
11
+
+110 o
PNN 11
BN are the mean and standard deviation of the Where L is the computed loss, dke and 1u o 1tth activation layer and the Batch-Normalization respectively.
By observing the sample visual representation of the retro-synthesit data comparing against the ran- dom data depicted in Fig. [ from the respective image classes in DNN understandable format. Hence using the retro-synthesis data for the estimation of activation ranges achieves better accuracy as compared to using random data. Also, outperforms the state-of-the
obvious that the retro-synthesis data captures relevant features
data-free quantization methods (Nagel et al. 2019 Cai et al.) (020) with a good accuracy margin when validated on models with and without Batch- Normalization layers. Therefore, the same data generation technique used the other two pro- posed quantization methods (b) and (c) as well.
3.2 HYBRID QUANTIZATION
In any quantization method, to map the range of floating-pon point values to integer values, parameters such scale and point needed. These parameters can be calculated either for per-layer of
4

---

Under review as a conference paper at ICLR 2021
Random ganseain data
Retre-synthesis
Figure 1:
A sample representation of the random Gaussian data and the Retro-synthesis data using the proposed in Algorithm respectively for sample class of ResNet-50 model.
the model or per-channel in each layer of the model. The former referred to as 'per-tensor/per- layer quantization' while the latter is referred to as 'per-channel quantization'. Per-channel quan- tization is preferred over per-tensor in many cases because is capable of handling the scenarios where weight distribution varies widely among different channels in a particular layer. However, the major drawback of this method is, it is not supported by all hardware (Nagel et al., (219) and also it needs to store scale and zero point parameters for every channel thus creating additional compu- tational and memory overhead. On the other hand, per-tensor quantization which is more hardware friendly suffers from significant accuracy loss, mainly at layers where the weight distribution varies significantly across the channels of the layer, and the same error will be further propagated down consecutive layers of the model resulting in increased accuracy degradation. In the majority of the cases, the number of such layers present in a model is very few, for example in the case of MobileNet- V2 only very few depthwise separable layers show significant weight variations across channels which result in huge accuracy loss (Nagel et al.) 2019). To compensate such accuracy loss per-channel quantization methods are preferred even though they are not hardware friendly and com- putationally expensive. Hence, in the proposed "Hybrid Quontization" " technique we determined the sensitivity of each layer corresponding to both per-channel and per-tensor quantization schemes and observe the loss behavior at different layers of the model. Thereby we identify the layers which are largely sensitive to per-tensor (which has significant loss of accuracy) and then quantize only these layers using the per-channel scheme while quantizing the remaining less sensitive layers with the per- per-tensor scheme. For the layer sensitivity estimation KL-divergence (KLD) is calculated between the outputs of the original FP32 model and the FP32 model wherein the i-th layer is quantized us- ing per-tensor and per-channel schemes. The computed layer sensitivity is then compared against threshold value (Th) in order to determine whether a layer is suitable be quantized using the per-tensor or per-channel scheme. This process is repeated for all the layers in the model.
The proposed Hybrid Quantization scheme can be utilized for a couple of benefits, one is for ac- curacy improvement and the other is for inference time optimization. For accuracy improvement, By doing this, a hybrid quantization model
the threshold value has be set to zero, Th with a unique combination of per-channel and per-tensor quantized layers is achieved such that, the accuracy is improved as compared to a fully per-channel quantized model and in some cases also FP32 model. For inference time optimization the threshold value Th is determined heuristically by observing the loss behavior of each layer that aims to generate a model with the hybrid approach, having most of the layers quantized with the per-tensor scheme and the remaining few sensitive layers quantized with the per-channel scheme. In other words, we try to create a hybrid quantized model close possible to the fully per-tensor quantized model so that the inference is faster with the constraint of accuracy being similar to the per-channel approach. This resulted in models where per-channel quantization is chosen for the layers which are very sensitive to per-tensor quantization. For instance, in case of ResNet-18 18 model, fully per-tensor quantization accuracy is 69.7% and fully per-channel accuracy is 71.48%. By performing the sensitivity analysis of each layer, we observe that only the second convolution layer is sensitive to per-tensor quantization because of the huge variation in weight distribution across channels of that layer. Hence, by applying per-channel quan- tization only to this layer and per-tensor quantization to all the other layers we achieved 10- 20% reduction in inference time, The proposed method is explained in detail in Algorithm 2 For every layer in the model, we find auxiliary model Auzzmodd/==qunntizzz (M, i, qs))) where, the step quantize(M, i, .s) quantizes the i-th layer of the model M using qs quant.scheme, scheme, where qs could be per-channel or per-tensor while keeping all other layers same the original FP32 weight val- ues. To find the sensitivity of layer, we find the KLD between the Aur-model and the original FP32 model outputs. If the sensitivity difference between per-channel and per-tensor is greater than
5

---

Under review as a conference paper at ICLR 2021
the threshold value Th, apply per-channel quantization that layer else apply per-tensor quantization. The empirical results with this method are detailed section 4.2
Algorithm 2 Hybrid Quantization scheme
Input: Fully trained FP32 model (M) with n layers, retro-synthesis data (X) generated in Part A. Output: Hybrid Quantized Model.
1. Init: schemee {PC, PT)
2. Init: Mg + +M
3. for i 0, 0, .2
(a) urror(PC]
error [P7]
(b) for (gs quant
i. Aux-model+- quantizo(M..i.qs)
ii Outputt- (X).
iii. Aux-outpute Aux-model(X)
iv. e KLD(Output, oux-uut
error [gs]
(c) if Meroo error PPC] < Th
Mg < quantize(MM i, PT)
else
M, + quantize(MMq. i, PC)
3.3 NON-UNIFORM QUANTIZATION
In the uniform quantization method, the first step is segregate the entire weights range of the given FP32 model into 2 groups of equal width, where 'K' is bit precision chosen for the quantization, like K = 8, 6, 4, etc. Since have total of 2 bins steps available for quantization, the weights in each group are assigned to a step or bin. The obvious downside with this approach is, even though, the number of weights present in each group is different, equal number of steps are assigned to each group. From the example weight distribution plot shown in Fig. 2 that the number of weights and their range in 'group-m' very dense and spread across the entire range, whereas they are very sparse and also concentrated within very specific range value in 'group-n'. In the uniform quantization approach since an equal number of steps are assigned each group, unfortunately, all the widely distributed weights in 'group-m' are quantized to a single value, whereas the sparse weights present in 'group-n' are also quantized to a single value. Hence it is not possible to accurately dequantize the weights in 'group-m', which leads to accuracy loss. Although a uniform quantization scheme seems to be a simpler approach is not optimal. possible scenario is described in Fig. 2 there may exist many such scenarios in real-time models. Also, in cases where the weight distribution has outliers, uniform quantization tends to perform badly as ends up in assigning reasonable to assign more steps to the groups with more number of weights and fewer steps to the groups with less number of weights. With this analogy, in the proposed Non-Uniform Quantization method, first the entire weights range is divided into three clusters using Interquartile Range (IQR) Outlier Detection Technique, and then assign a variable number of steps for each cluster of weights. Later, the quantization process for the weights present in each cluster is performed similar to the uniform quantization method, by considering the steps allocated for that respective cluster as the total number of steps.
it is evident
many steps even for groups with very few outlier weights. In such cases, it is
With extensive experiments, it observed that assigning the number of steps to a group by consid- ering just the number of weights present in the group, while ignoring the range, results in accuracy degradation, since there may be more number of weights a smaller range and vice versa. There- fore it is preferable to consider both the number of weights and the range of the group for assigning the number of steps for a particular group. The effectiveness of this proposed method is graphi- cally demonstrated for sample layer of the ResNet-13 18 model in Fig. Binn in the appendix A. By observing the three weight plots evident that the quantized weight distribution using the pro- posed Non-Uniform Quantization method is more identical FP32 distribution, unlike the uniform

---

FP32 Under review as a conference paper at ICLR 2021
1500
grogpe
gr9AP)
ranghtoragg
Figure 2: An example weight distribution plot of random layer in model with weight range of [0 - 3] divided into 6 groups of equal width. The ranges from 0.5 to 1 and 2.5 to 3 are labeled as 'group-m' and "group-n' respectively, For clarity only the weights in these two groups are shown.
quantization method and hence it achieves better quantized model. Also, it should be noted that the proposed Non-Uniform quantization method is fully per-tensor based method.
4 EXPERIMENTAL RESULTS
4.1 RESULTS FOR QUANTIZATION METHOD USING
DATA
Table [ shows the benefits quantization using the 'retro-synthesis' data 3.1 against state-of-the- art methods. In the case of models with Ratch-Normalization layers, the proposed method achieves 1.5% better accuracy against DFQ and a marginal improvement against ZeroQ. Also, our method outperformed FP32 accuracy in the case of ResNet-18 and ResNet-50., In the case of models without Batch-Normalization layers such as Alexnet and ISONet (Qi et al. 2020) the proposed method outperformed the ZeroQ method by 2 3% the ImageNet dataset.
Table 1: Quantization results using retro-synthesis data for models with and without Batch- Normalization layers on ImageNet dataset with weights and activations quantized to 8-bit (W8A8). BN field indicates whether the respective model has Natch-Normallzatioon layers present in it not.
Model
BN
DFQ
ZeroQ
Proposed method
FP32
resnet18
69.7
71.42
71.48
71.47
resnet50
77.67
77.67
77.74
77.72
mobilenetV2
V
71.2
72.91
72.94
73.03
Alexnet
X
55.91
56.39
56.55
ISONet-18
X
65.93
67.67
67.94
ISONet-34
X
67.60
69.91
70.45
1SONet-50
X
67.91
70.15
70.73
ISONet-101
X
67.52
69.87
70.38
Table 2 demonstrates the effectiveness of the proposed retro-synthesis data for low-precision (weights quantized to 6-bit and the activations quantized to 8-bit (W6A8)). From the results, evident that the proposed method outperformed the ZeroQ method.
Table 2: Results for quantization method using retro-synthesis data with weights quantized to 6-bit and activations quantized to 8-bit (W6A8) on ImageNet dataset
Model
ZeroQ
proposed method
ResNet-18
70.76
70.91
71.47
Resnet-50
77.22
77.30
77.72
MobileNet-V2
70.30
70.34
73.03
The efficiency of the proposed quantization method for lower bit precision on the CIFAR-10 dataset for ResNet-20 and ResNet-56 models is depicted in Table 3 below. From the results, it is evident

---

Under review as a conference paper at ICLR 2021
that the proposed method outperforms the state-of-the-et methods even for lower precision 8, 6, and bit weights with 8 bit activations.
Results for quantization method using retro-synthesis data with weights quantized to 8, 6, and 4-bit and activations quantized 8-bit (W8A8, W6A8, and W4A8) CIFAR-10 dataset
Table 3:
W8A8
W6A8
W4A8
Proposed method
Proposed method
Proposed method
Model
ZeroQ
ZeroQ
ZeroQ
ResNet-20
93.91
93.93
93.78
93.81
90.87
90.92
ResNet-56
95.27
95.44
95.20
95.34
93.09
93.13
4.2 RESULTS FOR HYBRID QUANTIZATION METHOD
Table 4 demonstrates the benefits of the proposed Hybrid Quantization method in two folds, one is for accuracy improvement and the other for the reduction in inference time. From the results, is observed that the accuracy is improved for all the models as compared to the per-channel scheme. should also be observed that the proposed method outperformed FP32 accuracy for ResNet- and ResNet-50. Also by applying the per-channel (PC) quantization scheme to very few sensitive layers as shown in "No. of PC layers" column of Table and applying the per-tensor (PT) scheme to remaining layers, the proposed method optimizes inference time by 10 20% while maintaining a very minimal accuracy degradation against the fully per-channel scheme.
Table 4: Results on ImageNet dataset with the proposed Hybrid quantization scheme
2 (W8A8),
with threshold Th 0 for accuracy improvement and set 0.001 for inference time benefit.
% time
Hybrid Th=0
Hybrid
No. of PC layers for Th=0.001
Model
PC
PT
FP32
benefit for Th=0.001
Th=0.001
Resnet-18 18
71.48
69.7
71.60
71.57
71.47
20.79
Resnet-50
77.74
77.1
77.77
77.46
77.72
2
17.60
MobilenetV2
72.94
71.2
72.95
72.77
73.03
4
8.44
4.3
RESULTS FOR NON-UNIFORM QUANTIZATION
Since the proposed Coi-oriform Quantization method is a fully per-tensor based method, quanti- tatively demonstrate its effect, we choose to compare the models quantized using this method against the fully per-tensor based uniform quantization method. The results with this approach depicted in Table 5 accuracy improvement of 1 is evident for the ResNet-18 model.
Table 5: Results with Non- Uniform Quantization method (W8A8) on ImageNet dataset
Uniform quantization 69.70
Model/Method
FP32
Quantization
ResNet-18 ResNet-50
70.60
71.47
77.1
77.30
77.72
5 CONCLUSION AND FUTURE SCOPE
This paper proposes data independent post training quantization scheme using "retro sysnthesis" data, that does not depend on the natch-Normalization layer statistics and outperforms the state-of- the-art methods in accuracy. Two futuristic post training quantization methods are also discussed namely "Hybrid Quantization" and Uniforn Quantization" which resulted in better accuracy and inference time compared to the ofate-of-the-ar methods. These two methods unleashes lot of scope for future research in similar lines. Also in future more experiments can be done on lower precision quantization such 6-bit, 4-bit and 2-bit precision using these proposed approaches.
8

---

Under review as a conference paper at ICLR 2021
REFERENCES
Ron Banner, Yury Nahshan, and Daniel Soudry. Post training 4-bit quantization of convolutional net- works for rapid-deployment. Advances in Neural Information Processing Systems, pp. 7950- 7958, 2019.
Chaim Baskin, Brian Chmiel, Evgenii Zheltonozhskii, Ron Banner, Alex M Bronstein, and Avi Mendelson. Cat: Compression-aware training for bandwidth reduction. arXiv preprint arXiv: 2019.
Yaohui Cai, Zhewei Yao, Zhen Dong, Amir Gholami, Michael W Mahoney, and Kurt Keutzer. Zeroq: A novel zero shot quantization framework. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13169-13178, 2020.
Yoni Choukroun, Eli Kravchik, Fan Yang, and Pavel Kisilev. Low-bit quantization of neural net- works for efficient inference. In 2019 JEEE/CVF International Conference on Computer Vision Workshop (ICCVW), pp. 3009-3018. IEEE, 2019.
Lucian Codrescu. Architecture of the hexagon TM 680 dsp for mobile imaging and computer vision. In 2015 IEEE Hot Chips 27 Symposium (HCS), pp. 1-26. IEEE, 2015.
Matthieu Courbariaux, Yoshua Bengio, and Pierre David. Binaryconneet: Training deep neural networks with binary weights during propagations. In Advances in neural information processing systems, pp. 3123-3131, 2015.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Fei-Fei. Imagenet: A large-scale hi- erarchical image database. In 2009 IEEE conference computer vision and pattern recognition, pp. 248-25822222 Icee, 2009.
Zhen Dong, Zhewei Yao, Amir Gholami, Michael W Mahoney, and Kurt Keutzer. Hawq: Hes- sian aware quantization of neural networks with mixed-precision. In Proceedings of the IEEE International Conference Computer Vision, pp. 293-302, 2019.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
Benoit Jacob, Skirmantas Kligys, Bo Chen, Menglong Zhu, Matthew Tang, Andrew Howard, Hartwig Adam, and Dmitry Kalenichenko. Quantization and training of neural networks for efficient sunngr--rithmmeti-onn inference. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2704-2713, 2018.
Raghuraman Krishnamoorthi. Quantizing deep convolutional networks for efficient inference: A whitepaper. arXiv preprint arXiv: 08342, 2018
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo- lutional neural networks. In Advances in neural information processing systems, pp. 1097-1105, 2012.
Rundong Li, Yan Wang. Feng Liang, Hongwei Qin, Junjie Yan, and Rui Fan. Fully quantized network for object detection. In Proceedings the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2810-2819, 2019.
Nelson Morgan et al. Experimental determination of precision requirements for back-propagation training of artificial neural networks. Proc. Second Int'l, Conf. Microelectronics for Neural Networks, pp. 9-16. Citeseer, 1991.
Markus Nagel, Mart van Baalen, Tijmen Blankevoort, and Max Welling. Data-free quantization through weight equalization and bias correction. In Proceedings of the IEEE International Con- ference on Computer Vision, pp. 1325-1334. 2019.
Haozhi Qi. Chong You, Xiaolong Wang. Yi Ma, and Jitendra Malik. Deep isometric learning for visual recognition. arXiv preprint arXiv:2006. 16992, 2020.

---

Under review as a conference paper at ICLR 2021
Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. In European conference on computer vision, pp. 525-542 Springer, 2016.
Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mo- bilenetv2: Inverted residuals and linear bottlenecks. Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4510-4520, 2018.
Jiaxiang Wu, Cong Leng. Yuhang Wang, Qinghao Hu, and Jian Cheng. Quantized convolutional neural networks for mobile devices. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4820-4825 2016.
Dongqing Zhang, Jiaolong Yang. Dongqiangzi Ye, and Gang Hua. Lq-nets: Learned quantization for highly accurate and compact deep neural networks. In Proceedings of the European conference on computer vision (ECCV), pp. 365-382, 2018.
Ritchie Zhao, Yuwei Hu, Jordan Dotzel, Christopher De Sa, and Zhiru Zhang. Improving neu- ral network quantization without retraining using outlier channel splitting. arXiv preprint 1rXiiv/901.09400 2019.
Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantiza- tion: Towards lossless cnns with low-precision weights. arXiv preprint 1rXiv:1702.03044, 2017.
Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint 16 2016
A APPENDIX
A.1 NON-UNIFORM QUANTIZATION METHOD
A.1.1 CLUSTERING MECHANISM
The IQR of range of values is defined as the difference between the third and first quartiles Q3, and Q1 respectively. Each quartile is the median of the data calculated as follows. Given, an even 2n or odd 2n+ + number of values, the first quartile Q1 is the median of the smallest values and the third quartile Q3 is the median of the largest values. The second quartile Q2 is same as the ordinary median. Outliers here are defined as the observations that fall below the range Q1 - 1.5IQR or above the range Q3 + 1. This approach results in grouping the values into three clusters C1, C2. and C3 with ranges R4 = [min, Q1 L.IQR), R2 = 1Q1 1.5QR, Q3 + 1.5QRR), and R3 (Q3 + 1.5QQR, max] respectively.
115
300
-05 4.14 -025 do Nap- Aal-Uniormn quantization distributia
Unifee guantlaation distribution
Figure 3: Weight distribution of FP32 model, model quantized using the proposed Non-Unifomm Quantization method and uniform quantization method in respective order for a sample layer in ResNet- 18 model. The horizontal and vertical axis represents the weights range and frequency respectively.
With extensive experiments it is observed that, assigning the number steps to a group by consid- ering just the number of weights present in the group, while ignoring the range, results in accuracy degradation, since there may be more number of weights a smaller range and vice versa. There- fore it is preferable to consider both number of weights and the range of the group for assigning the number steps for particular group. With this goal we arrived at the number of steps allocation methodology explained below in detail.
10

---

Under review as a conference paper at ICLR 2021
A.1.2 NUMBER OF STEPS ALLOCATION METHOD FOR EACH GROUP
Suppose W,, and RG represent the number of weights and the range of i-th cluster respectively, then the number of steps allocated S; for the i-th cluster is directly proportional to R& and W; as shown in equation below.
= (R₃ x Wi)
Thus, the number of steps S; allocated for i-th cluster can calculated from equation by deriving the proportionality constant C based the constraint E(S.) 2*, where A: is the quantization bit precision chosen. So, using this bin allocation method we assign the number of bins to each cluster.
Once the number of steps allocated for each cluster the quantization is performed on each cluster to obtain the quantized weights.
A.2 SENSITIVITY ANALYSIS FOR PER-TENSOR AND PER- FER-CHANNEL QUANTIZATION SCHEMES
0.005
per channel
per tensor
0.004
0.003
0.002
0.001
0.000
10
20
30
40
50
Sensitivity plot of per-tensor Vs per-channel quantization schemes for MobileNetV2 model
Figure 4: Sensitivity plot describing the respective layer's sensitivity for per-tensor and per-channel quantization schemes in case of MobileNet-V V22 model. The horizontal axis represent the layer num- ber and the vertical axis represents the sensitivity value.
From the sensitivity plot in Fig. sl it is very clear that only few layers in MobileNetV2 model are very sensitive for per-tensor scheme and other layers are equally sensitive to either of the schemes. Hence we can achieve betetr accuracy by just quantizing those few sensitive layers using per-channel scheme and remaining layers using per-tensor scheme.
A.3 SENSITIVITY ANALYSIS OF GROUND TRUTH DATA, RANDOM DATA AND THE PROPOSED SETRO-SYNTTESIS DATA
From the sensitivity plot inFig. 5 it is evident that there is clear match between the layer sensitiv- ity index plots of the proposed sttrosyyttheiis data (red-plot) and the ground truth data (green plot) whereas huge deviation is observed in case of random data (blue plot). Hence it can be concluded that the proposed retro-synthesis data generation scheme can generate data with similar characteris- ties as that of ground truth data and is more effective compared to random data.
11

---

Under review as a conference paper at ICLR 2021
0.0030
random data
retrosynthesis data
0.0025
Ground Truth
0.0020
0.0015
0.0010
0.0005
0.0000
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Sensitivity plot comparing Ground truth random data Vs retrosynthesis
Figure Sensitivity plot describing the respective layer's sensitivity for original ground truth dataset, random data and the proposed retro-synthesis data for ResNet-18 model quantized using per-channel scheme. The horizontal axis represent the layer number and the vertical axis represents the sensitivity value.
12


Reviews:
[{"review_id": "qD0YFASBjg7", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper considers the problem of data-free post-training quantization of classfication networks. It proposes three extensions of an existing framework ZeroQ (Cai et al., 2020): (1). in order to generate distilled data for network sensitivity analysis, the \"Retro Synthesis\" method is proposed to turn a random image into a one that represents a desired class label without relying on batch norm statistics like in ZeroQ; (2). a hybrid quantization strategy is proposed to optionally provide finer-grained per-channel quantization instead of the typical per-layer quantization; (3). a non-uniform quantization grid is proposed to better represent quantized weights, instead of uniform quantization as in ZeroQ.  Empirical evaluation demonstrate the effectiveness of the proposed approach.\n\n==========================================================\n\nPros:\n1. The proposed \"Retro Synthesis\" approach broadens the scope of ZeroQ to a wider range of neural network architectures by lifting the requirements of having batch normalization layers, and thus may significantly extend the practical applicability of data-free post-training quantization.\n2. The proposed approach seems to run very fast (but on the same order of magnitude as ZeroQ).\n\n\n==========================================================\n\nCons:\n1. The significance of the contribution appears limited in scope, as the three proposed methods (especially the latter two, \"hybrid quantization\" and non-uniform quantization) read more like incremental modifications to components of the existing ZeroQ framework. As far as I understood, the overall approach of this paper follows the ZeroQ framework, e.g., uses the generated distilled data in the same way, also computes the effect of quantization based on KL divergence, and allocates per-layer/channel bit precision using the same Pareto frontier approach. The proposed \"Retro Synthesis\" is the main innovation, but is rather poorly explained (what's the motivation for having three components L_BN, L_G, L_C, to the total loss, and are they really weighted equally in the experiments?) and lacks insight into why it works.  And it is questionable whether the remaining two proposals count as contributions: \"hybrid quantization\" is a simple heuristic for deciding when to use per-channel vs per-layer quantization, and non-uniform quantization is a slightly more sophisticated quantization grid (simply divides weights into quartiles and does uniform quantization in each quartile). I have no issue with a simple method if it's well motivated and works well, but:\n2. Moreover, the experiments are plagued with a lack of careful analysis of the proposed methods and details for reproducability. Since the three proposed improvements focus on different aspects of ZeroQ, ablations that analyze the effect of each in isolation  (i.e., baseline ZeroQ + proposed hybrid quantization, baseline ZeroQ + proposed non-uniform quantization) are crucial for evaluating their contributions. Details such as how the pre-trained models were obtained (were they trained from scratch, or pre-trained and obtained from the original ZeroQ authors) are not provided, which can significantly impact the empirical results and how they can be understood.  The issue is not specific to this paper -- the field of neural network compression can benefit from better reproducability and more robust evaluation, and particularly methods that operate post-training can evaluate on a shared repository of pre-trained models and report accuracy loss.\n\n==========================================================\n\nComments & Questions:\n1. Since the paper argues that \"it is possible to generate image data with similar statistics\" based on class scores (under the proposed \"Retro Synthesis\" approach), it makes sense to validate this is indeed the case qualitatively by comparing against the alternative \"ground truth\" approach based on matching batch norm statistics, e.g. by comparing the resulting model layer sensivity on distilled data produced by these two approaches, like in Figure 2 of the ZeroQ paper.\n2. What's the Gaussian loss (LG) component of the main loss function (eq (1))? Similarly, step (e) of algorithm 1 is unclear -- what is \\mu_0' and \\sigma_0'?\n3. Maybe I'm missing something, but wouldn't something obvious like k-means perform just as well (if not better) compared to the proposed non-uniform quantization approach based on uniformly quantizing within quartiles?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "RZFnmHmlFLO", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposed three different techniques to improve the quality of the post-training quantization (PT) results. The main claim is about Retro-Synthesis Data, which allows the calibration of the quantization parameters without the training data. There are two additional techniques, Hybrid Quantization and Non-Uniform Quantization, to increase the accuracy of PT. \n\nAlthough the overall paper is nicely written, the contents might need more novelty for being accepted in ICLR. Here are a few concerns related to it.\n\n- The \"Retro-Synthesis Data\" scheme sounds interesting, but not sure how much practical impact it might bring in. The authors assumed the model deployment scenario where the full-precision model is given without any training data. But it does not necessarily mean that the unlabeled data is also not available; in most cases, test data (without labels) from the deployed application is available for tuning of parameters (E.g., Sec 4 of [Sun et al., NeurIPS19]). In such a case, it is not clear how much benefit we can expect from \"synthesizing\" the test data for tuning quantization parameters.\n\n- Even if we believe that synthesizing the test data is necessary, it is not clear why \"Retro-Synthesis Data\" is well suited for the post-training quantization. Although the approach is interesting, how much representative the data created by the proposed method is for being used for the parameter tuning? It would be highly desirable to explain in the derivation of \"Retro-Synthesis Data\" how it can provide information particularly useful for better post-training quantization.\n\n- The strong empirical evidence might say more than just a detailed explanation. But unfortunately, the performance gain in the experimental results seems to be marginal. The authors claim that the proposed method is good for the models without batchnorm, but it does not sound very convincing given the fact that batchnorm is extremely popular these days...\n\n- The additional techniques, Hybrid Quantization and Non-Uniform Quantization, seem to be incremental from the previous work about quantization-sensitivity and outlier aware controls of neural networks (such as precision selection and channel splitting). Also, the authors claimed that there is 10~20% performance gain from Hybrid Quantization, but without much justification on how such hardware performance is measured/estimated.\n\nTo sum, this is a well-written paper but it might need more novelty for being accepted in ICLR. \n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "8-8sXqDjdHD", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work uses post-training quantization without access to training data for privacy concerns. Instead, useful statistics are estimated using a retro-synthesis data obtained from the FP baseline. I have a few comments, some concerns and some suggestions I think can be used to improve this work.\n\nOverview of prior work:\nMost competing approaches are included and representative prior arts are mentioned which is good. In the descussion about post-training quantization methods, the authors conclude that prior arts observe accuracy degradations. There have been works on post-training quantization that theoretically predict such degradation and overcome it by increasing precision as needed using the concept of noise gains [1]. I think the authors should contrast such work with theirs. It is my impression that noise gains in [1] can be used to improve the presented method.\n\nComments about the method:\nRetro-synthesis data generation:\nThe retro-synthesis data is used to determine activation ranges which are useful for quantization. This is a clever method. I wonder if that is really necessary though. Can't we follow an analysis similar to [2] in order to predict activation statistics from weight statistics (which are available)?\n\nHybrid quantization:\nIt seems the concept of per-tensor quantization has already been studied in [3] and also use the concept of noise gains as above to analytically determine the required number of bits for each tensor. I think this method can be useful to improve/validate the proposed hybrid quantization technique.\n\nNon-uniform quantization:\nI have two issues: (1) why not simply use the Lloyd-Max algorithm (which optimizes non-uniform quantization), (2) it is unclear how activations are quantized.\n\nThe experimental results look good.\n\n[1] Sakr, C., Kim, Y., & Shanbhag, N. Analytical guarantees on numerical precision of deep neural networks. In International Conference on Machine Learning, ICML 2017.\n\n[2] He, K., Zhang, X., Ren, S., & Sun, J. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, CVPR 2015.\n\n[3] Sakr, C., & Shanbhag, N. Per-tensor fixed-point quantization of the back-propagation algorithm. In 7th International Conference on Learning Representations, ICLR 2019.\n\nPost Rebuttal Comments:\n\nI thank the authors for their feedback. I have no modification to make to my original review.", "extra_scores": {}, "extra_text": {}}, {"review_id": "3QctDb4V5FS", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a data-independent post-training quantization scheme by generating a faux dataset without depending on the BN layer statistics of the FP32 model. The authors also introduce two variants of post-training quantization methods, hybrid quantization and non-uniform quantization methods. \n\nStrengths of the paper: \n- The proposed quantization method is practically applicable to privacy-constraint applications because no training data are required and the method works for any model. \n- The proposed Hybrid Quantization scheme addresses a couple of benefits, one is for improving accuracy and the other is for faster inference time. Both of them are important metrics for practical applicability.\n- The paper is well written and easy to follow.\n\nWeaknesses of the paper: \n- Novelty: The novelty is moderate but not strong enough. Specifically, (1) a zero-shot quantization framework to generate a synthetic dataset has been studied in the previous literature ZeroQ (using the distilled data engineered to match the statistics of BN layers to perform post-training quantization). But this paper employs an alternative faux dataset, called by the Retro-Synthesis Data, that does not depend on the BN layer statistics of the original data, which should be appreciated. Moreover, most of DNN models employ the BN layer in their construction, what is the merit in that case? (2) The same layer sensitivity metric as ZeroQ, the KLD between the original model and the quantized model has been used in the proposed Hybrid Quantization scheme. The main idea behind Hybrid Quantization using the KLD to determine whether a layer is suitable to be quantized using the per-tensor or per-channel scheme, seems to be same. Overall, the contribution is a bit incremental and seems not novel to me compared with ZeroQ and DFQ. \n- Evaluation: Further experiments and more ablation studies should be done. The provided experimental results are too weak to support the strength of using the retro-synthesis data.\n\nDetailed comments: \n\n(1) Algorithm 1 to generate retro-synthesis data seems to have a dependency on the target class(C). If there are many classes to be classified, is the time complexity also increasing? The paper states that generating the retro-synthesis data takes 10~12 sec. Is this for CIFAR-10 or ImageNet dataset? The proposed scheme seems to be more time-consuming than ZeroQ's 3 sec for generating the distilled data. In the ZeroQ paper, the end-to-end quantization of ResNet50 on ImageNet data takes almost 30 sec, which takes 3 sec to generate the synthetic data, 12 sec to estimate the sensitivity for all layers, and 14 sec to perform Pareto Frontier optimization etc. Could you provide the detailed timing breakdown on which experimental setting?\n\n(2) Computing the KLD in Algorithm 2 has more timing O/H than ZeroQ since it is executed twice for both per-tensor and per-channel quantization schemes per layer. Then does it take time twice more compared with ZeroQ? Could you provide the detailed timing breakdown or time O/H evaluation on your Hybrid Quantization configuration? Is the time O/H ignorable compared to the benefit of faster inference time by exploiting the hybrid quantization scheme? Approaches to achieve the goal of optimizing the inference time cause a conflict of interest.\n\n(3) A threshold value Th is sub-optimal and determined heuristically to decide either per-channel or per-tensor quantization. Is there any way to find an optimal value? The paper uses Th=0.001, but there is little improvement over Th=0. Could you explain why this happens? Further ablation study on different Th values should be provided. In the 8-bits setting, there seems no significant difference b/w per-channel and per-layer schemes due to its perturbation effect. More explanation on the result is required.\n\n(4) Results (in Table 3) of comparing the proposed method to ZeroQ with W8A8, W6A8, and W4A8 on ResNet model with CIFAR-10 are too weak to support the strength of using the retro-synthesis data. There seems to be a bit improvement over ZeroQ while the proposed scheme takes more time(10~12 sec for the retro-synthesis data) to generate a synthetic dataset compared with ZeroQ(3 sec for the distilled data).  \n\n(5) Results on ImageNet data (in Table 4 and 5) have a weak contribution if the bitwidth used in the experiment is 8 bit(You should describe a precision setting in the result tables). More aggressive setting of bitwidths, say 6 or 4 bits, should be provided comparing with the SOTA post-training quantization schemes. \n\n(6) The paper also presents another variant of using the retro-synthesis data, a non-uniform quantization scheme. Further evaluation (in Table 5) should be done to support its superiority by comparing it with the SOTA non-uniform quantization schemes, e.g. PoT, APoT, DDQ, etc. Especially, the non-uniform quantization gets more important in the weaker representation range, say low-precision quantization schemes below 8 bits. Future more experiments should be provided. \n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Four knowledgeable referees reviewed this paper; one reviewer (weakly) supports accept and other three indicate reject. Even with the rebuttal, all negative reviewers have concerns on the limited novelty and marginal performance improvement, and agree that the paper is not well qualified for the high standard of ICLR.
Decision: Reject