CBP-QSNN: Spiking Neural Networks Quantized Using Constrained Backpropagation
Authors: Donghyung Yoo
Doo Seok Jeong
Venue: Submitted to ICLR 2023 (2023)
Keywords: Quantized spiking neural network
Constrained backpropagation
Binary weight
Lagrange multiplier method
Weight constraint

Abstract: Spiking Neural Networks (SNNs) support sparse event-based data processing at high power efficiency when implemented in event-based neuromorphic processors. However, the limited on-chip memory capacity of neuromorphic processors strictly delimits the depth and width of SNNs implemented. A direct solution is the use of quantized SNNs (QSNNs) in place of SNNs with FP32 weights. To this end, we propose a method to quantize the weights using constrained backpropagation (CBP) with the Lagrangian function (conventional loss function plus well-defined weight-constraint functions) as an objective function. This work utilizes CBP as a post-training algorithm for deep SNNs pre-trained using various state-of-the-art methods including direct training (TSSL-BP, STBP, and surrogate gradient) and DNN-to-SNN conversion (SNN-Calibration), validating CBP as a general framework for QSNNs. CBP-QSNNs highlight their high accuracy insomuch as the degradation of accuracy on CIFAR-10, DVS128 Gesture, and CIFAR10-DVS in the worst case is less than 1\%. Particularly, CBP-QSNNs for SNN-Calibration-pretrained SNNs on CIFAR-100 highlight an unexpected large increase in accuracy by 3.72\% while using small weight-memory (3.5\% of the FP32 case).

Under review as a conference paper at ICLR 2023
CBP-QSNN: SPIKING NEURAL NETWORKS QUAN- TIZED USING CONSTRAINED BACKPROPAGATION
Anonymous authors
Paper under double-blind review
ABSTRACT
Spiking Neural Networks (SNNs) support sparse event- based data processing at high power efficiency when implemented in event-based neuromorphic proces- sors. However, the limited on-chip memory capacity of neuromorphic processors strictly delimits the depth and width SNNs implemented. direct solution is the use of quantized SNNs (QSNNs) in place of SNNs with FP32 weights. To this end, we propose a method to quantize the weights using constrained backpropagation (CBP) with the Lagrangian function (conventional loss function plus well-defined weight-constraint functions) as an objective function. This work utilizes CBP as post-training algorithm for deep SNNs preitrei using various state-of-the-ard methods including direct training (TSSL-BP, STBP, and surrogate gradient) and DNN-to-SNN conversion (SNN-Calibration), validating CBP general frame- work for QSNNs. CBP-QSNNs highlight their high accuracy insomuch as the degradation of accuracy on CIFAR-10. DVSI28 Gesture, and CIFAR IO-DS in the worst case is less than 1 1%. Particularly, CBP-QSNNs for pretrained SNNs on CIFAR- 100 highlight an unexpected large increase accu- racy by 3.72% while using small weight-memory (3.5% of the FP32 case).
INTRODUCTION
Spiking Neural Networks (SNNs) are time-depent deeeddeendeett models with spiking neurons whose dynamics in conjunction with synaptic current dynamics constitutes the rich dynamics of SNNs (Jeong) 2018).
Deep SNNs are clearly distinguished from deep neural networks (DNNs) such that (i) presynaptic spiking neurons send out 1-bit data (spikes a.k.a. events) to their postsynaptic neurons unlike the nodes sending out real-valued activation values to the nodes in the next layer in DNN and (ii) SNN operations are based on asynchronous sparse spikes unlike DNNs based on layerwise synchronous activation calculations (Jeong 2018; Pleifter & Přeil 2018). These distinct features endow SNNs with high power efficiency given minimum data movements and high sparsity in operations. Yet, SNNs leverage the efficiency only when implemented in neuromorphic processors that supports event-based operations.
Neuromorphic processor design technologies diverse, e.g., mixed analog/digital circuits (Merolla et 2014aa Moradi et al. 2018 Neckar et 20119), and fully digital circuits (Merolla et al. 2014b) Davies al. 20088: Frenkel et al.) 2018: Kornjeuu al. 2019). Albeit diverse, all designs com- monly suffer from their limited on-chip memory (SRAM) capacity. The on-chip memory is mainly assigned to neurons (state variables and hyperparameters), synapses (weights, state variables, and hyperparameters) and event-router (lookup tables). The largest portion of on-chip memory is ded- icated to synaptic weights given a significant number of synapses in a deep SNN. Additionally, most neuromorphic processors hardly allow weight-reuse for convolutional SNNs because they are designed for dense SNNs. Although some compilers for weight-reuse, e.g., NXTF for Loihi (Rueck- auer et al. 202I). the weight-reuse rate is still far below the ideal rate. Consequently, the limited on-chip memory capacity strictly limits the size (depth and width) of SNNs implementable neu- romorphic processors.
Considering the limitation of on-chip memory capacity, attempts to reduce the use of synaptic
weight-memory have been made, which include unstructured SNN pruning (Neftei al. 2016
Rathi et al., 2019; Martinelli et al.] 22000] Chen et al., 2021 Deng et al.) 2021; Kim et al.
2022
Chen et al.
2022) and weight-quantization (Rueckauer et al. 2017 Yousefzadeh et al. 2018
Srini-

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CIFAR-10
CIFAR-100
ImageNet
92
70
66
SNN Call CBP
SNIV-Coll + CBP
Accuracy (%)
91
66
Weng al. (2020)
BS CBP
lu et al. (2020)
90
TP
SNN-Cal CBP
62
Deng (2021)
SEW RosNet:18 CBP
SEW ResNet:18 CBP
SNN-Cal CBP
58
SEW ResNets
CBP
89
Wang et al. (2020)
58
SEW.RasNet:34 CBP
Dang al. (2020)
Lu (2020)
Zhou (2021)
Zhou al. (2020)
880
20
40
60
80 100 120
54
20
40
60
80
100
120
54
50
100
150
200
250
Weight memory usage (Mb)
Weight memory usage (Mb)
Weight memory usage (Mb)
Figure 1: Accuracy and weight-memory usage of CBP-QSNNs on CIFAR-10/100 and ImageNet. The diamond, square, and star symbols denote binary, ternary, and int2 weight precision, respec- tively.
vasan & Roy 2019 Lu & Sengupta 2020 Kim et al. 2021 Deng et al.) 2021 Eshraghian et al.) 2022). Although unstructured SNN pruning can increase the sparsity of weight matrices, it hardly reduces the memory usage unless low precision data formats are used for such null weights, which, however, causes additional significant complexity in computation. Weight-quantiation to lower resolution than the full-precision (FP32) can largely reduce weight-memont usage. Several meth- ods proposed to date, however, are limited to (i) particular learning algorithms, e.g., spike timing- dependent plasticity (STDP) (Yousefzadeh et al. 2018 Srriivasan & Roy, (019), event- bised ran- dom backprop (eRBP) (Kim et al. [2021), rate code-based spatio-temporal backprop (Deng et al.) 2021), or (ii) image datasets (e.g., MNIST, FashionMNIST, CIFAR-10/100 ImageNet) for the con- 2017 Wang et al. 2020
version of quantized DNNs to quantized SNNs (QSNNs) (Rueckauer et al.
Lu & Sengupta, 2020).
Generally, deep SNNs learn optimal weights using three distinct methods: direct training using
backprop based on (i) rate code (Wu et al. 2018 Shhrestha & Orchard, 2018 Wu et al., 2019 Fang
ct al.
2021a b Zheng et al. 2021) and (ii) temporal code (Zhang & Li| 2020 Yang et al. 2021
Zhou et al. 2021). J and (iii) DNN-to-SN conversion (Rueckauer et al. |217 Sengupta al.
2019
Han et al. 2020: Deng & Gu 2020 Li et al. 2021). Note that the last method covers SNNs on im-
age (static rather than based- datasets only. Regarding this diversity in learning methods and
datasets, a general nright-quantization framework can resolve the difficulty in
To this end, we propose a weight-quantization method based on constrained backpropagation (CBP) that uses Lagrangian functions (conventional loss function plus weight conht-cnnnntainn functions, each of which given a I Lagrange multiplier) as objective functions (Kim & Jeong 202T). CBP offers a gen- eral weight-quantization framework given that any constraints on weight quantization (e.g., binary, ternary, etc.) can easily be applied to the weight-constraint functions in the Lagrangian function. CBP is post-training method so that SNNs that learned real-valued weights using different pre- training methods can be post-trained using the Lagrangian functions with the same loss functions as for the pre- training. The main contributions of our work
follows:
We validate CBP general framework for QSNNs by successfully binary- and ternary- quantizing deep SNNs (with various topologies) pre-trained on various datasets (CIFAR- 10/100, ImageNet, DVS128 Gesture, CIFARIO-DVS) using representative (i) rate code- based backprop algorithms with surrogate gradients (Wu et al. 2018 Fang et al. aa) (ii) temporal code-based backprop algorithm (Zhang & Li 2020). and (ii) DNN-1o-SN conversion (Li et al. 2021).
We propose surrogate loss function quantize SNNs that learned real-valued weights using the DNN-to-SNN conversion method and report surprisingly high accuracy, partic- ularly, on CIFAR-100, which exceeds the accuracy of the real-valued SNNs by more than 3%.
We analyse the accuracy and weight-memory efficiency of CBP-QSNNs on various datasets in comparison with previous methods, which highlights the state-of-the th--t---tttttttt (SOTA) accu- racy with weight-memory usage similar to or lower than the previous methods as shown in Figure I
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2
RELATED WORK
Simple methods for weight-quantization include learning uniformly quantized weights using straight through estimators (STEs) with surrogate gradients (Zhou et al. 2021 Qiao et al. 2021 Eshraghian et al. 2022). This quantization method inevitably causes large noise that hinders the loss from attaining its minima. Zhou et al. (2021) overcame the difficulty by using a novel neuron model with differential phase-domain integrator, whereas Eshraghian et al. (2222 overcame by periodically boosting the learning rate.
Several methods convert binarized DNNs QSNNs have been proposed. [Rueckauer et al.. (2017) proposed a method to convert BinneyConnect (Courbariaux et al. (2015) and binarized neural net- works (Courbariaux et al. 2016) to binary QSNNs, whereas Lu & Sengupta (2020) binary weight networks (Rastegari et al. 2016) to binary QSNNs. Unlike direct training methods, such conversion methods do not suffer from the issue arising from -infeefferee spiking functions, but such binary SNNs are incapable of processing event- datasets.
There exist weight-quantization methods tailored to particular learning algorithms. Yousefzadeh et al. (2018) proposed the probabilistic update of binary weights using STDP for shallow dense SNNs. Srinivasan & Royl (2019) introduced ReStoCNet with one convolutional layer with binary kernels and one dense layer; the convolutional layer is trained using probabilistic STDP model with binary weights, whereas the dense layer using backprop. These quantization methods suffer from the inherent drawback of STDP, i.e.. its limited scalability, and thus limited learning capacity. Kim et al. (2021) proposed the eWB algorithm based on the Lagrangian function as an objective function, which tailored to eRBP. Given the limited scalability of eRBP, eWB is also limited to shallow SNNs on datasets of low complexity. Deng et al.] (2021) proposed an augmented La- grangian funetion-based weight-quantization method, where the augmented Lagrangian function is minimized using the alternating direction method of multiplier (ADMM) (Boyd et al. 2011). This method is tailored to STBP with power-of-two and zero weights.
3 METHOD
3.1
CONSTRAINED BACKPROPAGATION
Objective function Constrained backpropagation (CBP) is post-training method to apply in- dependent arbitrary constraints to each weight by training DNNs using Lagrangian functions L as objective functions (Kim & Jeong, 202T). For CBP-QSNNs, we define the Lagrangian function consisting of a conventional loss functions C and weight-constraint function cs as follows.
E(0, W A) = C(O, W) A cs()),
cs (W) [es (w)) cs( " and A (AI....,ANN AN.,)
J(=1 where O is a set of SNN outputs all timesteps, O [o and Ô is the desired output. For temporal codes, Ô is also given by a set of desired outputs all timesteps, Ô {o') { i.e., desired output spike sequence, whereas, for rate codes, it is frequently given by a one-hot vector that indicates the correct label. Given that the loss function C' in Equation ( should be identical that for pre-training, the choice of types of O and Ô relies on the pre-training method. The total number of weights in the SNN and the set of all N weights are denoted by No, and W, respectively. The value X for i (1.
No is the Lagrange multiplier for weight w/:-
Constraint function In this work, we consider constraints of binary weights Q = (-1,1} and ternary weights Q (, 1). To this end, choose sawtooth-shape constraint functions es that attain their minima (cs = 0) at q E Q. but modified by inconstrained-weigge windows ucs in which the equation es holds. The width of ucs(w) is parameterized by g(> 1) which is shared among all weights.
Ng n1-1
ucs (w) = -1- Σ H
(9++1 q:) - - m, + cc)
2g
i=1
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(a)
(b)
Constraint function
Constraint function
Constraint function
Constraint function
2
9=1
9=3
9=1
9=3
1
1
1
0
0
-2
-1 1
2
1
-2
-1
0
1
-2
0
2
Weight
Weight
Weight
Weight
Figure 2: Constraint functions with unconstrainee--weight windows (filled areas) for (a) binary- and (b) weiiht constraints.
where na = QQ), and q and 0+1(> 9:) are neighboring elements Q. The median of d and 9--1 is denoted by m (q: + 9++1/ and e - 0+. The function H denotes the Heaviside step function. When = 1, ucs = 0 for all w E [min Q. max Q]. The width of ucs narrows down as g increases, and it vanishes when 9 + oc. Examples of binary- and ternary-wett weight constraint functions for and 3 are shown in Figure
Parameter update CBP includes three types of parameters subject to update: W, A, and g. The weights W and multipliers À are updated using the Lagrange multiplier method (LMM). LMM with differentiable constraint functions cs calculates the optimal W and A such that (W*, X*) = arg minw. L (W, A). To include constraint functions that are con-differentiable at W apply pseudo-LMM (Kim & Jeong, 202I) to Equation ①.. which is given by
V A) =0 0 { C (W) 0, cs (W)
0.
That is, at the minima of the Lagrangian function L, the loss function attains its minimum, and the optimal W satisfies the constrains. In search of W we use the basic differential multiplier method (BDMM) (Platt & Barr 1987p. which updates W and À using a gradient descent and ascent method, respectively. Note that the weight update conforms to the backpropgation pipeline for the pre-training. but with the Lagrangian function L rather than the loss function C.
W +
W 7/ w C (W. (W,)),
(2)
where f/ denotes the learning rate for the weights W.
Unlike the weight update, the multipliers À and unconstrained-weiihh window parameter are up- dated once every learning epoch only if the the sum of the Lagrangian functions over the mini- batches for given epoch (Lsum) exceeds the sum for the previous epoch (C or (ii) the update has not been executed in the past pmna epochs.
{g
+ A (W, A) = X + 7ACS (Wig). + + --++Ag where Ag = (if < 10) and 10 (if g 2 10).
where 7/x denotes the learning rate for the Lagrange multipliers A. The variable g is initialized to and conditionally increases (never decreases) during training. Thus, the unconstrained-wweipe weight window ucs allows rather gradual quantization of weights such that the weights outside the window (i.e., close to q E Q) are forced be quantized (which likely increase the loss) while the weights within the window (far from Q) are updated to reduce the loss. CBP is elaborated in Algorithm in Appendix A.1
Application CBP can works general weight-quantizatie framework because the algorithm itself is tailored to neither particular weight constraints nor particular loss functions. The loss func- tion C' in Equation (I) and backpropagation pipeline for weight update can readily be chosen as per the loss function used for the pre- pre-training. The weight-constraing
can freely be chosen as per the weight resolution desired. In the following two sections, will show applications of CBP to eral distinct cases for which pre- pre-trate trained SNNs with FP32 weights and/or codes are publicly available for fair baselines.
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3.2 CBP SETTINGS FOR DIRECT TRAINED SNNS
TS conv-SNNs (temporal code) TSSL-BP is renowned temporal code-based error- emor-backpropagaatoon algorithm with data labels encoded as distinct output spike sequences O
{or) 7 o' = s', where s' denotes the desired output spike vector at timestep t) Zhang & Li 2020 Likewise, the output of the SNN O is sequence of output spikes over the timesteps, O = { where s denotes the actual output spike vector at timestep t. The loss function C is defined using the van Rossum distance
C(O,0) Σ
(e 8') (e (f)))))'
(3)
where c is an exponential kernel. CBP seamlessly applies to TSSL-BP by plugging Equation into Equation ( The weights are updated while being quantized conforming Equation (2). To this end, the backpropagation pipeline for TSSL-BP is used with the same hyper-parameters. TSSL- BP bypasses spike function gradients (which is non-differentiable) in its backward pass: instead, it addresses the gradient potential with respect to presynaptic spike timing, which is available in closed-form (Zhang & Li 2020).
STBP-trained conv-SNNs (rate code) STBP is a representative backpropagation algorithm for STBP takes the time-averaged number of output spikes as
SNNs with rate code (Wu et al. 2018). the output in response input sample; O 1/T -
s The label for given sample (Ò) is encoded as a one-hot vector. The loss function for STBP in (Wu 2018) is given by
1 Ô - 01 2
C
STBP includes the gradients of spike functions s with respect potential (0s/Ou) du) in its backprop- agation pipeline, which are approximated using boxcar functions which are centered at the spiking threshold & (Wu et al. 2018).
Ōs Du
{
Iu 0l a. otherwise,
where a is a positive constant.
BP-trained SEW-ResNet (rate code) SEW-ResNet is version of ResNet, which consists of residual blocks with residual connections for identity mapping (Fang et al.] (2021a). The keys to SEW-ResNet are the configuration of computational blocks and introduction of suitable spike- element-wise logic functions to fit the identity mapping in SNNs. Fang et al.) (2021a) deployed integrate-and-firg neurons in the output layer and considered their potential (u²) which is normalized by the the number of total timesteps T, i.e., a = u" /T, as the output of the SNN. The output vector then processed by the subsequent softmax layer use the crossentropy loss as the loss function C.
the last timestep
C(O,O)= -
Σ
de log (0c),
(4)
0. exp(ac)
u
where ag
Ic=1 exp(ac)
T
SEW-ResNet also involves the gradients of spike functions with potential u, which are approximated to the arctan-based surrogate gradient function (Fang et al. 202
as
(5)
Du
1 + (aT (u 0))²
where is positive constant.
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3.3 CBP SETTINGS FOR SNNS CONVERTED FROM DNNS
DNN-to-SNN conversion methods often apply to indirectly train SNNs (Rueckauer al. 2017
Sengupta et al. 2019 Han et al. 2020: Deng & Gu 2020 Li et al. 2021). Among them, we chose the SNN-Calibration algorithm (Li al. 2222 given its SOTA accuracy on various static datasets. SNN-Calibration minimizes accuracy-degradation for the limited number of timesteps by attaining the optimal layer-wise threshold (g(1) for a layer 1) that minimizes the discrepancy in layerwise output between the SNN and its DNN counterpart. Additionally, (i) bias for each neuron and/or (ii) initial potentials and weights are tweaked to minimize the layerwise output error.
The WNN-CClibration-i -reettiinnd- SNN utilizes spiking neurons that output real-valued spikes ((),
which is not suitable for CBP. Therefore, we first rescale the weights W biases ( and threshold g(t) in the pre-trained SNN as follows.
4l, W t w(() 00-1)
009 (
000 t 1.
(6)
(0))
This rescaling allows the spiking neuron output integer binary spikes.
Given that conversion methods do not use loss functions, we employ surrogate loss functions C that are consistent with the neural code used in the pre-trained SNN. SNN-Calibration uses the time-averaged number of spikes (spike-rate) as neural input and output, so that rate code-based loss functions are appropriate surrogate loss functions as for the previous two examples. Here, we chose a cross-entropy loss function with a softmax layer as a surrogate loss function C as for SEW- in Equation (4). Similar to SEW-ResNet, we also used the arctan surrogate gradient function in Equation (5) with
1. However, not only weights W but also biases b were subject to post- learning. Note that the biases b were updated using the loss function as an objective function using the learning rate (= nw). Stochastic gradient descent was used to learn optimal biases b* as for the weights. CBP for this case is detailed in the pseudocode in Algorithm 2 in Appendix(A.I
4 EXPERIMENTS
We identified the classification efficacy and weight-memory efficiency of CBP-QSNN: (with bi- nary Q = {-1, 1} and ternary weight Q = 0, constraints) for (i) TSSL-BP-trained conv-SNNs, (ii) STBP-trained conv-SNNs, (iii) BP-trained SEW-ResNet, and (iv) SNN-Calibration- trained SNNs with the settings elaborated in Secion 3 We used total five datasets: three image datasets (CIFAR-10/100 (Krizhessky 2009) and ImageNet (Russakovsky
al. 2015)) and two event-based datasets (DVS 128 Gesture (Amir et al. 2017) and CIFARIO-DVSA et al. 2017)). Note that we used publicly available pre-trained models and/or codes only for fair baselines. The input pre- pe-pprccessing and encoding conformed to the methods for the pre-training without modi- fications, which are detailed in the corresponding publications. CBP needs two optimizers (one for weight update and the other for Lagrange multiplier update). For the weight update, we used the same optimizer as for the weight pre-training, which are listed in Table 4 in Appendix A.2 Hereafter, we refer CBP- QSNN for SNN pre-traiee pre-trained using particular algorithm CBP-QSNN. prre-training algorithm), c.g., CBP-QSNN-(TSS) (TSSL-BB)) and CBP-QSNN-(SNN- CEN-SSNN-SNN-CCaibbttionn except SEP-QSNN- ResNet) whose pre-training algorithm does not have a compact name.
We used layerwise scaling factors c()) for layers I, which are given by c()) W (l//n() where
( and (1))) are the weight matrix for the layer and the number of elements in (l)), respec- tively. For CBP-QSNN-(TSSS-BB) we used Adam W (Loshchilov & Hutter 2019) Lagrange multiplier optimizer, whereas Adam Kingma & Ba (0144) for the others. The hyper-papaaee parameters for CBP-QSNNs are listed in Table in Appendix A.2 Note that the weights for the first and last layers
were not subject to quantization as for common QDNNs (Rastegari et al. 2016 Li et al. 2016). All statistic data were acquired from three trials to avoid the random seed effect except ImageNet. The CBP algorithm was implemented in Pytorch (Paszke et al. 2019) on a GPU workstation (RTX A6000).

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Table 1: Classification accuracy and weight-memory usage for CBP-QSNNs that were pre-trained using direct training methods.
Network
Algorithm
Weight precision
Avg. (Best) accuracy
weight-memory usage
CIFAR-10
TS
FP32
89.00 229)
677Mb
AlexNet
Binary
89.45-10.37 (89.95)
21.6Mb
CBP
Ternary
88.92:10.03 (88.96)
42.7Mb
STBP
FP32
90.26' (89.531)
1.99Gb
7Conv, 3FC
CBP
Ternary
Binary
89.58 (89.67)
-0.055 (89.59)
62.4Mb
124Mb
CIFAR-100
STBP
FP32
62.25
1.99Gh
7Conv, 3FC
Binary
60.81 (60.83)
63,8
CBP
Ternary
60.85-10.18 (61.09)
126Mb
ImageNet
Arctan gradient
FP32
62.81
374Mb
SFW-ResNet-Ie
CBP
Ternary
Binary
54.34
58.04
39.0Mb
27.8Mb
Aretan gradient
FP32
66.651 (67.041)
697Mb
SEW-ResNet-34
Binary
60.10
37.9Mb
Termary
62.98
59.2Mb
DVS128 Gesture
Arctan gradient
FP32
(97.99
4.16Mb
7B-Net
Binary
97.11=#0.33 (97.57)
159Kb
CBP
Ternary
97.68 (97.92)
288Kb
CIEARIO-DYS
Arclan grudient
FP32
(74.44)4
40.2Mb
Wide-7B-Net
CBP
Ternary
Binary
74.83:10.46 (75.3)
7..89::0.62 (74.7)
2.58Mb
1.33Mb
Reproduued result Reported result
To measure the success in weight quantization, we introduce the constraint- score (CFS)
which is the average weight- function over all weights included in the SNN.
1 N.. CFS(W) = N
cs(w:),
where No indicates the total number of weights in the SNN. During quantization, CFS keeps de- creasing toward zero because the constraint functions es attain their minima (cs = 0) when the weights fully quantized. Few weights that have not been quantized until the last CBP epoch were forced to be quantized to their nearest quantized weights.
4.1 CBP-QSNNS WITH TRUE LOSS FUNCTIONS
CBP-OSNN-(TSSL-BBP) We applied CBP with binary- and ternary-weight constraints AlexNet 2AAE://P22.BBACAP2222232222333-0
pre-trained on CIFAR- 10 using TSSL-BP. The results shown in Table I highlight surprising increase classification accu- racy by 0.45% when binarized, using merely a weight-memory of 21.6Mb, i.c., 3.19% of that for FP32 weight case. The test accuracy and CFS curves are plotted in Figure 3 in Appendix A.3 Additionally, the weight distributions sampled at few epochs are plotted in Figure 3 which show gradual
CBP-OSNN-STBE We considered the deep SNN (7Conv,3FC) (Deng et al. 2021) which was pre-trained on CIFAR- 10 using STBP, yielding an accuracy of 90.26 with FP32 weights (Table IT]. The corresponding CBP-QSN 1BP-QSNN-STBB- with binary- and cemmryywwiig constraints show accuracy degradations of 0.75% and 0.68% while reducing weight- usage by 96.9% and 93.8%, re- spectively. The test accuracy and CFS curves, and weight distributions sampled at four learning

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Table 2: Classification accuracy and weight-memory usage for CHP-OSNN--SNN-Callraann
Network
Algorithm
Weight precision
Avg. (Best) accuracy
weight-memory usage
CIFAR-10
SNN-Calibration
FP32
91.04 (91.821)
471Mb
VGG-16
Binary
9..40-0.09 (91.51)
15.1Mb
CBP
Ternary
91.49 (91.66)
29.8Mb
CIFAR-100
SNN-Calibration
FP32
62,60 (64.534)
473Mb
VGG-16
CBP
Binary
66.3 (66.53)
16.6Mb
Ternary
6..47:10.44 (66.46)
31.2Mb
Reproduced result *Reported result
epochs are shown in Figure 4 in Appendix A.3
Despite the non-availability of baseline perfor-
mance in public, we also applied CBP to the same SNN on CIFAR-100 100 for comparison with another weight-quantization method, which will be addressed in Section 4.3) The results are also listed Table I and the quantization behaviors are plotted Figure Appendix A.3
We addressed the pre-trained NEW-ResNet-18 18 and SEW-ResNet-34 on ImageNet, which use ADD operations as element-wise operations in the SEW blocks. CBP applied to SEW-ResNet-18 and SEW-ResNet-33 with binary- and ternary-weight constraints yielded the accuracy and weight-memory usage shown in Table T The quantization curves are plotted in Figures and and
in Appendix(A.3 A.3
We also applied CBP to 7B-Net and Wide-7B-Net pre-trained on neuromorphic datasets (DVS128 Gesture and CIFARIO-DVS, respectively). 7B-Net is structured 32C3-BN-PLIF- (SEW (bock(32C)-MP2)+7-FF0
where BN denotes batch normalization. PLIF denotes layer of parametric LIF neurons which use trainable membrane time constants (Fang et al. 20216). Wide-7B-Net is structured as 64C3-BN-PLIF- (4Gcc(64C)-MP2)) (4CC)-MMP)2411803333-3 11 (SWW
Block(128))MMP2) F3-FCO.. The results are shown in Table and the quantization curves are plot- ted in Figures 9in in Appendix A.3 An anomaly is seen in the ternary-weight CBt.-weeighCCBQSSSMMM whose
accuracy is rather larger than the FP32 counterpart by 0.13%.
4.2 CBP-QSNNS WITH SURROGATE LOSS FUNCTIONS
TTT-OONNN-ONN-Calbbbaan
VGG-16 SNNs were CBP-quantized on CIFAR-10 and 100
with binary- and ternary-weigh constraints. To acquire -BP-OSSWNSNN--Ciibrriiin,, we used the surrogate loss function and gradient function as for CBP- OSB- and (5). respectively. We applied weight decay (L2-regularization) with a decay rate wa of 10 T 2isows the accuracy and weight-
in Equations (4)
usage, highlighting considerable increases in accu- racy upon quantization by 0.36% (binary case) and 0.45% (ternary case) on CIFAR-10, and 3.72% (binary case) and 3.77% (ternary case) on CIFAR-100 100. The quantization curves and sampled weight distributions on CIFAR- 10 and 100 are plotted in Figures 10 and 11 in Appendix A.3 respectively.
4.3 COMPARISON WITH PREVIOUS QUANTIZATION RESULTS
We finally compare CBP-QSNNs with QSNNs by previous weight-quantization methods in terms of classification accuracy and weight-memory usage on CIFAR- 10/100, ImageNet, DVS128 Gesture, and CIFARIO-DVS. The comparison is shown in Table
CIFAR-10: CBP-QSNNs exhibit outstanding accuracy and weight meightymemory efficiency. Particularly, CBP-QSNN-STBR with ternary weights outperforms (Deng et al. 202T) of the same network archi- tecture by 0.57% in accuracy. Additionally, the ternary TEB-OSSN--SNNNCalibbll
of VGG-16 exceeds the SOTA accuracy of (Zhou et al. 2021) by 0.56% with reduction in weight-memory usage by 72.7%.
of VGG-16 outperforms the SOTA accuracy (Wang cet al. 2020) by 4.3% by using an additional 3.6Mb weight-memory only. CBP-QSNN- C--N--NNN---PPP of
CIFAR R-FAR- CBP-QSNN
conv-SNN (7Conv,FFC) records 60.85% in accuracy, exceeding the accuracy of (Deng et al. 2021)
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Table 3: Comparison with previous works QSNNs with binary, ternary, and int2 weight precision.
Algorithm
Network
Weight precision memory usage (bit)
Accuracy (%)
CIFAR-10
SNN RRecckaer
6Conv,
Binary/14.0M
BinaryConnecr Konss se 2017
6Conv, 3FC
Binary/14.0M
83,35
ConnerCooneet-Sii SNN Waanere ee al 2020
6Conv, 3FC
1inary/14.0M
90.19
STBP + ADMM (Deig er 2024
7Conv, 3PC
lemarv/12MM
89.01
Direct-training STE (hou
2021
SpGlingVGGI6
int2/109M
90.93
CBP-QSNN-STBP
7Conv,
Binary/62-4M
89.51 0.05 (89.59)
Termary/124M
89.58 0.08 (89.67)
CHP-QSNN.(TSSL.BB)
AlexNet
Binary/21.6W
(89.95)
Termary/42.7M
88.92 0.03 (88.96)
CBP-OSSN--ANN-Callrrtton
VGG-16
Termary/29.8M
Binary/15. IM
91.49=00 0.13 (91.66)
91.40-0.09 (91.5))
CIFAR-100
BinaryConnect SNN Wane et al. 2020
6Cpnv, 2PC
3.00
62,02
BWN-SNN (Lu & Scfrunie NDA0
VGG-15
Binary/32.11 1M.
62.00
STBP + ADRIVT (Deng 2021
7Conv, 3FC
1126
55.95
CHP-QSNN-STBP
7Conv, 3FC
Ternary/126
Binary663.8M
641.85 0..18 (61.09)
60.81-0.00 (60.83)
CHR-QSSw.-SNN-Caibrraii
VGG-16
Termary31.22
Binary/16.6M
66.37 0.08 (66.46)
(66.53)
ImageNet
BWN-SNN Sengunta 2010
VGG-15 GoogLeNet
Binary249M
62.71
Direct training STE Zhau Caa 2021
in1244.5M
60.0
Binary/27.8
SEW-ResNet18
Ternary/39.0
58.04
CHP-OSNN-SSEV-ReNNe
SEW-ResNet34
Ternary/59.2M
Binary/37.9M 9M
60.10
DVS128 Gesture
BSNN Qiao al. 2021
2Conv, 2FC
Binary/L.68M
97.57
CBP-QSNN-(TB-Nct
7B-Net
Ternary/288K
Binary/159K
07.6 (97.92)
97.11=0.33 (97.57)
CIFARIO-DYS
BSNN Qiao et al.
2021
2Conv, 2FC
Binary/L.6MM
62.10
Wide-7B-Net
Ternary/2.58M
Bin
73.89-14.62 (74.7)
74.83-1-.46 (75.3)
25% comnection-prune.
of the same network topology and weight precision by 4.9%. Note that the accuracy in (Deng et al. 2021] is of a 25% connection-prunee SNN.
ImageNet: CBP-QSNNN-SEWWRRSt
with ternary weights outperforms the SOTA results Lu & Sengupta 2020) by 0.27% in accuracy with reducing weight-memory usage by 76.2%.
DVS128 Gesture: For DVS 128 Gesture, CBP-Q CBP-QSNN-(7B-Net with ternary weights highlight an accuracy improvement by 0. % and a weight-memory reduction by 82.9% compared with QQaa et al. (02II.
with binary weights achieves a considerable accuracy improvement by 11.79% with reduction in weight-memory usage by 20.8% compared with (Qiao et al. 2021).
CIFARIO-DVS:
CONCLUSION
We proposed CBP for weight-quantization in SNNs, which can readily be adjusted to particularly pre-trained SNNs and desired weight- qugntiiquanizztioonn conditions. We validated CBP as general frame- work for QSNNs by reporting total 18 weight-quantizatton cases that vary in weight bitwidth (binary and ternary), pre-training method (TSSL-BP, STBP, arctan gradient, and SNN-Calibration) network topology, and dataset (CIFRA-10/100 ImageNet, DVS128 Gesture, and CIFAR 10-DVS). CBP for all datasets highlights its SOTA results terms of classification accuracy and niggt-mmemory usage in comparison with previous methods for QSNNs. Therefore, CBP-QSNNs high accuracy and high memory- -eemory-effciiet models that are mapped onto a neuromorphic processor with strictly limited on-chip memory capacity.

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Under review as a conference paper at ICLR 2023
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A APPENDIX
A.1 PSEUDOCODE
Algorithm 1: CBP algorithm. N denotes the number of training epochs in aggregate. M denotes the number of mini-batches of the training set Tr. The function minibatch( samples a mini- batch of training data and their targets from Tr. The function SNN returns output O for æ.
Input: Pre- tre-trained weights W in 32FP
Output: Quantized weights W, ?
Initialization such that À +- 0,p +- 0.g t 1; Initial update of X;
for epoch = 1 to do
Ssum -- 0:
/* Update of weight W
for i to M do
x a (
minibatch(TTr::
+ SN W);
L + C( W) X2 cs (W:9);
Leum + Leum + C:
W + clip(W lJ L)
end
Update of window variable g and Lagrange multiplier */
p p 11
if Lm
2 L or Pmax then
9 f g + Ag:
X < À cs (Wig);
p - 0;
C.re
maxx
else
Cree Lsumi
end
end
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Algorithm 2: CBP for SNN-Calibration
Input: Pre-trained weights W and biases b in FP32
Output: Quantized weights W and optimal biases b" Initialization such that VI, w g(t) /000, 600) + 1000///0 000 1;
Initialization such that À t 0, t 0, g 1 Initial update of X; for epoch = to N do
Lumm <-
/* Update of weights W and biases b to M do
for (6) - miri
O + SNN(x("); W):
+ C(O.Ó; W b.b X cs (W:g): Leum + Leum +
W +- CW Ij Vw
b - clip(b 1h VCC;;
end
/* Update of window variable g and Lagrange multiplier X */
p - p
Camm 2 Lo or = Pmax then
9 9 + Ag:
CA (W:g):
p 0;
Ceee fmax.
else
Cum f- Laumi
end
end
A.2 HYPER-PARAMETERS FOR CBP
Table 4: Hyper-parameters used in our experiments. The learning rates for weights W, Lagrange multipliers X, and biases b are denoted by Yor, and Tho respectively. The weight-decay (L2 regularization) is denoted by wd.
Optimizer weight
Optimizer multiplier
Algorithm
Dataset
7p
Thn
71x
Batch size
Epoch
Steps
TSSL-BP
CIFAR-10
2e.4
1e.4
50
200
5
Adam
Adam
CIFAR-10 10 CIFAR-100
STRP
5e-3
5e-4
SGD
Adam
DVS128 Gesture CIFARIO-DVS
16
SEW-ResNet
0.1
0.01
16
64
16
SGD
Adam
ImageNet
CIFAR-10
SNN-Calib
CIFAR-100
0.01
lo-4
0.01
1e-3
128
100
32
SGD
Adam
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A.3
EXTRA DATA
Binary weight constraint
10
90
1st
Test acc (%)
CFS (x1E-2)
50th
4
100th
Fraction
10-2
150th
85
10
80
10-6
100
200
100
200
Epoch
Epoch
Normalized weight
Ternary weight constraint
6
100
Test acc (%)
CFS (x1E-2)
50th
10
100th
Fraction
150th
85
10-4 -
80
0
10-6
100
200
0
100
200
-2
-1
1
Epoch
Epoch
Normalized weight
Figure 3: CBP-OSNN-CTSSL-BP) on CIFAR-10. The QSNN topology is of AlexNet.
Binary weight constraint
10
90
1st
Test acc(%)
30th
70
CFS(x1E-3)
4
60th
Fraction
10-2
90th
50
10
30
0
10-6
50 Epoch
100
50
100
Epoch
Normalized weight
Ternary weight constraint
90
6
100
1st
Test acc(%)
CFS(x1E-3)
30th
60th
Fraction
10-2
90th
50
10-4
30
10-6
50
100
0
50
100
2
-1
1
Epoch
Epoch
Normalized weight
Figure 4: CBP-QSNN-STEE CIFAR-10. The QSNN topology is of deep SNN (7Conv and 3FC).
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Binary weight constraint
70
6
100
1st
Test acc(%)
30th
50
CFS(x1E-3)
60th
Fraction
10-2
90th
10
10
0
10-6
50 Epoch
100
0
50
100
2
-1
1
Epoch
Normalized weight
Ternary weight constraint
70
100
1st
Test acc(%)
50
CFS(x1E-3)
30th
60th
Fraction
10-2
90th
30
2
10-4
10
50 Epoch
100
0
50
100
-2 -1
1
2
Epoch
Normalized weight
Figure 5: CBP-QSNN-STBP on CIFAR-100, The QSNN topology is of a deep SNN (7Conv and 3FC).
Binary weight constraint
90
10
12
1st
Test acc (%)
CFS (x1E-2)
32th
70
64th
Fraction
10-2 -
96th
50
10-4
Top-1
Top-5
50
100
50
100
2
-1
0
1
2
Epoch
Epoch
Normalized weight
Ternary weight constraint
10
12
1st
Test acc (%)
CFS (x1E-2)
32th
64th
Fraction
10-2
96th
10-4
Top-1
Top-5
10-6
50
100
50
100
-1
Epoch
Epoch
Normalized weight
Figure 6: CBP-QSVN--SFV-ReNNee111 on ImageNet.
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Binary weight constraint
90
12
10
1st
Test acc (%)
CF5 (x1E-2)
32th
70
64th
Fraction
10-2
96th
50
4-
10
Top-1
Top-5
30
0
10
50
100
0
50
100
-2
-1
1
2
Epoch
Epoch
Normalized weight
Ternary weight constraint
90
12
10
1st
Test acc (%)
CFS (x1E-2)
32th
70
8
64th
Fraction
10-2
96th
4-
10
Top-1
Top-5
10-6
50
100
50 Epoch
100
-2
-1
1
2
Epoch
Normalized weight
Figure 7: CHP-QSNN--SFV-eesNee..3)) on ImageNet.
Binary weight constraint
100
10
1st
Test acc (%)
95
CFS (x1E-2)
80th
160th
Fraction
10-2
M
90
10-4
85
80
10-6
0
100
200
0
100
200
-2
-1
0
1
2
Epoch
Epoch
Normalized weight
Ternary weight constraint
100
10a
1st
Test acc (%)
95
CFS (x1E-2)
80th
160th
Fraction
10-2
WWM
90
10
85
80
10-6
100
200
0
100
200
-
-1
2
Epoch
Epoch
Normalized weight
Figure 8: CBP-QSNN-(7B-Ne) on DVS128 Gesture.
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Binary weight constraint
76
100
1st
Test acc (%)
20th
Mudw M PAM
40th
Fraction
10-2
72
60th
CFS
10
70
10
30 Epoch
60
30
60
-2
-1
1
2
Epoch
Normalized weight
Ternary weight constraint
76
10
1st
Test acc (%)
74
MyMM M
CFS (x1E-2)
3
20th
Fraction
10-2
40th
60th
10
70
10-6
30 Epoch
60
30
60
-2
-1
1
2
Epoch
Normalized weight
Figure 9: CHP-QSWN-Wid-=7B-NNa) on CIBARIO-DVS.
Binary weight constraint
100
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Figure 10: CBP-OSNNN(SSW:ccllbaaaann) on CIFAR-10. The QSNN topology of VGG-16.
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Under review as a conference paper at ICLR 2023
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Figure 11: CBP-OSNNN((NNNcclbbrriion) on CIFAR-100. The QSNN topology is of VGG-
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Reviews:
[{"review_id": "zajT9DbqGDC", "reviewer": "Reviewer_6mKQ", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "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": "The manuscript makes a case for weight quantisation being necessary in deep SNNs.  It is shown that basic quantisation can introduce noise.  The authors then describe a constrained backprop using Lagrangian functions as objectives, showing that these can be tuned to different architectures.  Experiments appear to show that the method works both absolutely and when compared to other quantisation methods.", "summary_of_the_review": "The manuscript presents what appears to be a genuinely useful technique.  However, the theoretical justification is quite involved and the results are difficult to read.\nWhilst the material is strictly in scope, my feeling is that the manuscript in general is on the edge of the technical remit for *this conference*; this is evidenced to some extent by there being only one reference to past ICLR events.\nOn balance, I have too many questions to err more positively.", "strength_and_weaknesses": "Strengths:\nThe method seems to work.  It is applicable to more or less any kind of SNN irrespective of how it was trained.\n\nWeaknesses:\nIt is not clear to me quite how beneficial the method is given the tables.  The tables are quite difficult to read and do not always appear to compare like with like.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "NChVIk98XY", "reviewer": "Reviewer_4gmV", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "This paper presented CBP-QSNN, which applies the CBP algorithm proposed by Kim & Jeong in a recent study of quantized DNN as a general post-training method to quantized SNN. The authors began by introducing key elements of CBP, including the Lagrangian objective function and the weight-constraint function cs(w) in it, and pseudo-LMM aiming to include non-differentiable situation. Specifically, sawtooth-shaped cs(w) is combined with an unconstrained-weight window to make weights closer to the quantized value move toward this value preferentially. Then different CBP settings for various training methods and network structures of SNN were discussed to validate that CBP can work as a general weight-quantization framework for SNN. Finally, the authors reported experiments comparing the accuracy and weight memory usage, showing that CBP can significantly reduce the weight-memory usage without reducing performance or even improving performance. However, in the experiments comparing the CBP method with the previous works on QSNN, most comparisons are invalid because of the different network structures used. ", "summary_of_the_review": "This paper applied the CBP algorithm to SNN weight quantization and validated that CBP can be used as a general post-training framework for weight quantization by testing on various datasets with different SNN training methods. However, whether it is DNN or SNN, there is not much difference between the two for the CBP algorithm, so the innovation of this paper is insufficient. And there is a problem with the experimental setup in the article. When comparing CBP-QSNN with previous QSNN methods, there is no guarantee that the network structure is the same, which makes the experimental results unconvincing. And the performance improvement in the results compared with other algorithms is not attractive enough.", "strength_and_weaknesses": "Strength\n\n1. This work applied CBP to QSNN and achieved good performance in various datasets using several representative learning algorithms of SNN. Even if FP32 weights are quantized to binary or ternary weights, the test accuracy of CBP-QSNN can still be basically maintained or even better. This can greatly reduce memory consumption when deploying on neuromorphic chips while maintaining performance, thereby allowing deeper and wider SNN to be tried.\n\n2. CBP has been applied on different training methods for SNN (rate code-based backprop, temporal code-based backprop and DNN-to-SNN conversion), validating it to be a general framework for QSNNs.\n\n\nWeaknesses\n\n1. In this paper, the DNN weight quantization algorithm CBP is applied in SNN weight quantization. However, after the application is changed from QDNN to QSNN, the procedures of the algorithm actually have no change at all. Although the limited memory of neuromorphic chips makes SNN weight quantization more important, the work in this paper is still not innovative enough.\n\n2. The design of experiments to compare with previous works on QSNNs (Table 3) is unreasonable, because there is no guarantee that the variable is unique. For example, BSNN has different network structure from CBP-QSNN-(7B-Net), then the comparison between them cannot tell whose quantization algorithm is better. In the few comparisons with the same network structure, the advantages of the CBP algorithm are not obvious. For example, CBP-QSNN-STBP with ternary weights outperforms (Deng et al., 2021) by only 0.57% in accuracy. Moreover, the experimental results on ImageNet (table 1,3) are relatively poor and I don’t know why only ImageNet isn’t tested three trials to avoid the random seed effect.\n\n3. Some parts of the text are not clearly described: the equation of the weight-constraint function cs(w), the equation of the surrogate loss function proposed by authors for weight quantization of DNN-to-SNN conversion, calculation method of weight-memory usage, what does the clip function in the pseudocode do, and so on.\n\n4. I don't understand why the curve of test accuracy in A.3 (Figure. 3-11) always starts from a very low value, shouldn't the model has already been pre-trained when the post-training starts?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "-94tp-TjeYC", "reviewer": "Reviewer_k52R", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "The author proposes a method to quantize the weights of SNN model using constrained backpropagation (CBP) with the Lagrangian function as an objective function. CBP achieves SOTA results in terms of classification accuracy and weight-memory usage in comparison with previous methods for QSNNs on CIFRA-10/100, ImageNet, DVS128 Gesture, and CIFAR10-DVS datasets.", "summary_of_the_review": "See above\n", "strength_and_weaknesses": "Strength: \n\nthe paper is clearly written and easy to follow, figure 1 directly conveys the main results and the contribution of this paper.\n\nWeaknesses:\n\n(1) The related work is narrowed to the SNN quantization area, but there are many papers for quantization DNNs, such as BinaryConnect, BNN, XNOR[1], TWN[2]. The author has mentioned BinaryConnect, BNN in the related work and said that \"such\nbinary SNNs are incapable of processing event-based datasets\".  However, the reviewer is confused that why CBP is suitable for event-based datasets and prior works are not. Since the paper focus on the quantization of weights, please note why previous DNN quantization methods can not be applied to SNN weights. \n\n(2) The constraint functions are quite heuristic. \n\nFor example, in figure2, how are the functions designed? for figure 2(a) with g=1, are weights larger than 1 stored as fp32? The reviewer thinks that even if the weight matrix is quantized with mostly 0 value and very sparse fp32 values, the position/index of fp32 values will occupy additional memory. \n\n(3) The experimental results seem quite mixed, which are inconsistent with the SOTA claims in the conclusion. \n\nFor example, in Table3 CIFAR10, CBP-QSNN-STBP and STBP + ADMM have the same network architectures and similar accuracy (from 89.01 to 89.67 is no big deal for cifar10), but the memory usages are also the same, which make the reviewer very confused since the main goal of this paper is to reduce memory.\n\nFor CIFAR-10, CBP-QSNN-STBP with 63.8M weights performs worse than BinaryConnect-SNN with only 13.0M weights. \n\nThe authors are encouraged to re-organize the experimental results, with the same network arch and similar # weights, CBP has better accuracy, or with the same network arch and similar accuracy, CBP has less memory usage, etc.\n\n\n\n[1] Rastegari M, Ordonez V, Redmon J, et al. Xnor-net: Imagenet classification using binary convolutional neural networks[C]//European conference on computer vision. Springer, Cham, 2016: 525-542.\n\n[2] Li F, Zhang B, Liu B. Ternary weight networks[J]. arXiv preprint arXiv:1605.04711, 2016.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject