Optimal ANN-SNN Conversion for High-accuracy and Ultra-low-latency Spiking Neural Networks
Authors: Tong Bu
Wei Fang
Jianhao Ding
PENGLIN DAI
Zhaofei Yu
Tiejun Huang
Venue: ICLR 2022 Poster (2022)
Keywords: Spiking Neural Networks
ANN-SNN Conversion
Ultra-low Latency
Quantization Clip-floor-shift Activation

Abstract: Spiking Neural Networks (SNNs) have gained great attraction due to their distinctive properties of low power consumption and fast inference on neuromorphic hardware. As the most effective method to get deep SNNs, ANN-SNN conversion has achieved comparable performance as ANNs on large-scale datasets. Despite this, it requires long time-steps to match the firing rates of SNNs to the activation of ANNs. As a result, the converted SNN suffers severe performance degradation problems with short time-steps, which hamper the practical application of SNNs. In this paper, we theoretically analyze ANN-SNN conversion error and derive the estimated activation function of SNNs. Then we propose the quantization clip-floor-shift activation function to replace the ReLU activation function in source ANNs, which can better approximate the activation function of SNNs. We prove that the expected conversion error between SNNs and ANNs is zero, enabling us to achieve high-accuracy and ultra-low-latency SNNs. We evaluate our method on CIFAR-10/100 and ImageNet datasets, and show that it outperforms the state-of-the-art ANN-SNN and directly trained SNNs in both accuracy and time-steps. To the best of our knowledge, this is the first time to explore high-performance ANN-SNN conversion with ultra-low latency (4 time-steps). Code is available at https://github.com/putshua/SNN_conversion_QCFS

Published as a conference paper at ICLR 2022
OPTIMAL ANN-SNN CONVERSION FOR HIGH- ACCURACY AND ULTRA-LOW-LATENCY SPIKING
NEURAL NETWORKS
Tong Bu', Wei Jianhao Ding', PengL.in Dai², Zhaofei Yu Tiejun Huang!
Peeknng University. Southwest Jiaotong University
"Corresponding author: yuzf12@pku.eduu pku.edu.cn
ABSTRACT
Spiking Neural Networks (SNNs) have gained great attraction due to their dis- tinctive properties of low power consumption and fast inference on neuromorphic hardware. As the most effective method to get deep SNNs, ANN-SNN conversion has achieved comparable performance as ANNs on large-scale datasets. Despite this, it requires long time-steps to match the firing rates of SNNs to the activation ANNs. As a result, the converted SNN suffers severe performance degradation problems with short time-steps, which hamper the practical application of SNNs. In this paper, we theoretically analyze ANN-SNN conversion error and derive the estimated activation function of SNNs. Then we propose the quantization clip- floor-shift activation function to replace the ReLU activation function in source ANNs, which can better approximate the activation function of SNNs. We prove that the expected conversion error between SNNs and ANNS is zero, enabling us achieve high-accuracy and ultra-low-latency SNNs. We evaluate our method on CIFAR-10/100 1001000/00000000 and ImageNet datasets, and show that it outperforms the state- of-the-art ANN-SNN and directly trained SNNs in both accuracy and sime-steps. To the best of our knowledge, this is the first time to explore high-performance ANN-SNN conversion with ultra-low latency (4 timessteps). Code is available at wttp:///wwwaoooooouuuuuSoSSNn...
INTRODUCTION
Spiking neural networks (SNNs) are biologically plausible neural networks based on the dynamic characteristic of biological neurons (McCulloch Pitts] 1943], Izhikevich] (000). As the third gen- eration of artificial neural networks (Maass 1997). SNNs have attracted great attention due to their distinctive properties over deep analog neural networks (ANNs) (Roy et al.] 2019). Each neuron transmits discrete spikes to convey information when exceeding a threshold. For most SNNs, the spiking neurons will accumulate the current of the last layer as the output within inference time steps. The binarized activation has rendered dedicated hardware of neuromorphic computing (Pei) 2019
et 2019 DeBole et al.) 2019 Davies et al.) 2018). This kind of hardware has excellent advantages temporal resolution and energy budget. Existing work has shown the potential of tremendous en- ergy saving with considerably fast inference (Stöckl & Maass 2021).
In addition to efficiency advantages, the learning algorithm of SNNs has been improved by leaps and bounds in recent years. The performance of SNNs trained by backpropagation through time and ANN-SNN conversion techniques has gradually been comparable
ANNs on large-scale datasets (Fang et al.) 2021 Rueckauer et al. (017)). Both techniques benefit from the setting of SNN inference time. Setting longer time-steps in backpropagation can make the gradient of surro- gate functions more reliable (Wu et al. 2018 Neftci et al.) 2019 Zenke & Vogels 2021). However, the price is enormous resource consumption during training. Existing platforms such as TensorFlow and PyTorch based on CUDA have limited optimization for SNN training. In contrast, ANN-SNN conversion usually depends on a longer inference time to get comparable accuracy as the original ANN (Sengupta et al. 2019) because it is based on the equivalence of ReLU activation and integrate- and-fire model's 's firing rate (Cao et al.) 2015). Although longer inference time can further reduce the conversion error, it also hampers the practical application of SNNs neuromorphic chips.

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The dilemma of ANN-SNN conversion is that there exists a remaining potential in the conversion theory, which is hard to be climinated in a few time steps (Rueckauer et al.) 2016). Although many methods have been proposed to improve the conversion accuracy, such weight normaliza- tion (Diehl et al. 2013), threshold rescaling (Sengupta et al. 2019). soft-reset (Han & Roy, (220) and threshold shift (Deng & Gu 2020), tens to hundreds of time-steps in the baseline works are still unbearable. To obtain high-performance SNNs with ultra-low latency (e.g., 4 time-stess, we list the critical errors ANN-SNN conversion and provide solutions for each error. Our main contributions are summarized as follows:
We go deeper into the errors in the ANN-SNN conversion and ascribe them to clipping error, quantization error, and unevenness eтror. We find that unevenness error, which caused by the changes in the timing of arrival spikes and has been neglected in previous works, can induce more spikes or fewer spikes as expected.
We propose the quantization clip- floor-shift activation function replace the ReLU activa- tion function in source ANNs, which better approximates the activation function of SNNs. We prove that the expected conversion error between SNNs and ANNs is zero, indicating that we can achieve high-performance converted SNN at ultra-low time-steps.
We evaluate our method on CIFAR-10, 10, CIFAR-100, and ImageNet datasets. Compared with both ANN-SNN conversion and backpropagation training methods, the proposed method exceeds state-of-the-arr accuracy with fewer time-steps. For example, we reach top-1 1 accuracy 91. 11% on CIFAR-10 10 with unprecedented 2 time-steps.
2 PRELIMINARIES
In this section, we first briefly review the neuron models for SNNs and ANNs. Then we introduce the basic framework for ANN-SNN conversion.
Neuron model for ANNs. For ANNs, the computations of analog neurons can be simplified as the combination of a linear transformation and a non-linear mapping:
a' h( h(
I 1,2,
where the vector a' denotes the output of all neurons 1-th layer, Wi denotes the weight matrix between layer I and layer 1 - 1, and h(-) is the ReLU activation function.
Neuron model for SNNs. Similar to the previous works (Cao et al., 2015) Diehl et al. 2015 Han et al. 2020)) we consider the -negrate-and-Fire (IF) model for SNNs. If the IF neurons in 1-th layer receive the input -1) from last layer, the temporal potential of the IF neurons can be defined as:
m'(t) d' - 1) + W x t'))-
where m (1) and (t) represent the membrane potential before and after the trigger of spike at time-step t. W denote the weight in I-th layer. As the firing threshold 8', the neuron will elicit a spike and update the membrane potential (t). To avoid information loss, we use the "reset-by-subbractton" mechanism (Rueckauer et al. 2017 Han
as any element (t) of m mt(t) exceeds
eet al. 2020) instead of the "reset-to-zero" mechanism, which means the membrane potential (t) is subtracted by the threshold value 0¹ if the neuron fires. Based on the threshold-triggered firing of the membrane potential after firing discussed above,
mechanism and the rreset-by-subtracttin can write the uplate rule of membrane potential as:
s' (t) = Hm (t) - -').
(3)
e et) m (t) s' (t)g'.
(4)
Here st(t) refers to the output spikes of all neurons in layer at time the element of which equals if there is a spike and 0 otherwise. H--) is the Heaviside step function. Ø' the vector of the firing threshold 0'. Similar to Deng & Gu (2020), we suppose that the postsynaptic neuron in I-th layer receives unweighted postsynaptic potential 01 if the presynaptic neuron in ! 1-th layer fires a spike, that is:
x x1(() s' (t)6².
(5)
2

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Table 1: Summary of notations in this paper
Symbol
Definition
Symbol
Definition
Layer index
x (t)
Unweighted PSP
Neuron index
(())
Output spikes
Weight
< (T)
Average unweigthed PSP before time
a'
ANN activation values Time-stops
z
Weighted input from I 1 layer
k(-)
ReLU function
T
Total timc-step
H(-
Hcaviside step function
Ø'
Threshold
L
Quantization step for ANN
Trainable threshold ANN
Err'
Conversion Error
m(())
Potential before firing
Err
Estimated conversion Error
(t())
Potential after firing
4
Shift of quantization clip-floor function
Postsynaptic potential
ANN-SNN conversion. The key idea of ANN-SNN conversion is to map the activation value of an analog neuron in ANN to the firing rate (or average postsynaptic potential) of a spiking neuron in SNN. Specifically, we can get the potential update equation by combining Equation Equation
(t) o' 1) w'
s' (t)6'.
Equation 6 describes the basic function of spiking neurons used in ANN-SNN conversion. By summing Equation 6 from time to and dividing on both sides, we have:
(T) (0)
W' ii=1 x (i) Σ s'())'
T
T
T
we use (T- from O to T and substitute Equation
denote the average postsynaptic potential during the period 5 into Equation 7 then we get:
o' (T) (0)
D'(T) a'
T
Equation 8 describes the relationship of the average postsynaptic potential of neurons in adjacent layers. Note that Øf (T) O. If we set the initial potential (0) to zero and neglect the remaining term v'(T) when the simulation time-steps T is long enough, the converted SNN has nearly the same activation function as source ANN (Equation [T. However, high 7 would cause long inference latency that hampers the practical application of SNNs. Therefore, this paper aims implement high-perffrmance ANN-SNN conversion with extremely low latency.
3 CONVERSION ERROR ANALYSIS
In this section, we will analyze the conversion error between the source ANN and the converted SNN in each layer in detail. In the following, we assume that both ANN and SNN receive the same input from the layer simplicity, we use z' W' (T) = w'a to substitute the weighted input from layer - 1 for both ANN and SNN. The absolute conversion error is exactly the outputs from converted SNN subtract the outputs from ANN:
1, that is, a (T), and then analyze the error in layer l. For
Err' (B((T) a'=='' v (T)
(T)) v' h(2²),
where l(22) = (z) can be found from Equation that the conversion error is nonzero o' BT) 2 # 0 and 0. In fact, the conversion enror caused by three factors.
Clipping error. The output Φ (T) of SNNs is in the range of [0, as ('(T) = 2 3'
o gl (see Equation However, the output a' of ANNs much lager range of [0, a where a denotes the maximum value of a'. As illustrated in Figure [ a' can be mapped to d'(T (T) by the following equation:
0¹'
a' x
('(T) = clip
00,0'
(10)
3

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(a) Clipping error
(b) Even spikes
(c) More spikes
(d) Fewer spikes
Figure 1:
Conversion error between source ANN and converted SNN. 81 and 2/2 denote the output spikes of two neurons layer - 1, and of denotes the output spikes of a neuron in layer
Here the clip function sets the upper bound 0' and the lower bound 0. [-] denotes the floor function.
X' represents the actual maximum value of output a' mapped to the maximum value d of ø' (T)).
Considering that nearly 99.9% activations of a' in ANN in the range of [0, Ornaa Rueckauer al. 2016) suggested choose X' according 99.9% activations. The activations between X and a amaz in ANN are mapped to the same value 6 in SNN, which will cause conversion error called clipping error.
Quantization error (flooring error). The output spikes st) are discrete events, thus (T)) are discrete with quantization resolution (see Equation 10p. When mapping a' to e'' there exists unavoidable quantization error. For example, as illustrated in Figure Db, the activations of ANN in the range of / 22') are mapped to the same value o of SNN.
Unevenness error. Unevenness error is caused by the unevenness of input spikes. If the timing of arrival spikes changes, the output firing rates may change, which causes conversion error. There two situations: more spikes as expected or fewer spikes as expected. To see this, in source ANN, we suppose that two analog neurons in layer
are connected to an analog neuron in layer I with weights 2 and -2, and the output vector -1 of neurons in layer | - is [0.6, 0.4]. Besides, in converted SNN, we suppose that the two spiking neurons in layer 7 fire spikes and spikes in
Thus, " (T) -و '(6)
time-steps (T=5), respectively, and the threshold g- [0.6,0 0.4]. Even though (T' al-1 and the weights are same for the ANN and SNN, (T(T) can be different from a' if the timing of arrival spikes changes. According to Equation [ the ANN [0.6, 0.4]² = 0.4. As for SNN, supposing that the threshold 0' =,
output a' = l = [2. | there are three possible output firing rates, which are illustrated in Figure [ (b)-(d). If the two presynaptic neurons fires at t 1, 3, and t = 2, 4 (red bars) respectively with weights and -2, the
postsynaptic neuron will fire two spikes / 1,3 (red bars), and Φ (T) = مو = 0.4 a'. However, if the presynaptic neurons fires at = 2,3 and L = 4, 5, respectively, the postsynaptic neuron will fire four spikes at t = 2, 3, 4, and Φ' (T) = 0.8 > a'. the presynaptic neurons fires 1, 2, respectively, the postsynaptic neuron will fire only one spikes at f 5, and Φ (T) = 0.2 < a'
at t = 3, 4, 5 and
Note that the clipping error and quantization error have been proposed in et (2021) There exist interdependence between the above three kinds of errors. Specifically, the unevenness error will degenerate the quantization error if o' (T) is in the range of [0, 8"). Assuming that the potential v' (T) falls into [0, 0 will enable us to estimate the activation function of SNNs ignoring the effect of unevenness error. Therefore, estimation of the output value &' (T) in converted SNN be formulated with the combination of clip function and floor function, that is:
IT + v' (0)
(T) D'(T)
o
0 clip
(11)
The detailed derivation is in the Appendix. With the help of this estimation for the SNN output, the estimated conversion error Err can be derived from Equation
'
v((
Err 0l clip
.0, h1(22) Err
(12)
8
4

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023
(a) L = =4
(b) L 4,7
(c) 4,7
0.5
Figure 2: Comparison of SNN output (T) and ANN output a' with same input z
4 OPTIMAL ANN-SNN CONVERSION
4.1 QUANTIZATION CLIP-FLOOR ACTIVATION FUNCTION
According to the conversion error of Equation 12 it is natural to think that if the commonly used ReLU activation function h(2²) is substituted by clip-floor function with a given quantization steps L, (similar to Equation TTp. the conversion error at time-steps 7 L. will be eliminated. Thus
the performance degradation problem at low latency will be solved. As shown in Equation 13 we proposed the quantization clip-floor activation function train ANNs.
( Z'L X 0.1),
a' = h(2)) = clip
(13)
where the hyperparameter L denotes quantization steps of ANNs, the trainable X' decides the maximum value of a' in ANNs mapped to the maximum of (T) in SNNs. Note that zt W (T) W'a--1 (-1, With this new activation function, we can prove that the estimated con- version error between SNNs and ANNs is zero, and we have the following Theorem.
Theorem 1. An ANN with activation function
13 is converted to SNN with the same weights. If
8¹ X and u (0) then:
Err = (T(T) a' =0
(14)
Proof. According Equation 12 and the conditions L, 0¹ X1 r e'(0) 0, we have Err
o (T) a' 0¹ clip + x T-o (0)
.0.1) X' clip ( Z 0. 0.
Theorem 1 implies that if the time-steps T of the converted SNN is the same as the quantization steps L. of the source ANN, the conversion error will be zero. example is illustrated Figure Zh. where 7 = 1, 4, 0 = X The red curve presents the estimated output ø' p'(T) of the converted SNNs with respective to different input z', while the green curve represents the out a' of the source ANN with respective to different input z'. As the two curve are the same, the estimated conversion
error Err is zero. Nevertheless, in practical application, at different time-steps. There no guarantee that the conversion error is zero when T is not equal to L. As illustrated in Figure 2b. where L 4 and L 8, greater than zero for some z'. This error will transmit layer-by-layer and eventually degrading the accuracy of the converted SNN. One way solve this problem is train multiple source ANNs with different quantization steps, then convert them to SNNs with different time-steps, but comes considerable cost. In the next section, we propose the quantization clip-floor activation function with a shift term to solve this problem. Such an approach can achieve high accuracy for different time-steps, without extra computation cost.
focus on the performance of SNNs
find the conversion error is
4.2 QUANTIZATION CLIP-FLOOR SHIFT ACTIVATION FUNCTION
We propose the quantization flop-florr-hhift activation function train ANNs.
Z
a' = hh X' clip
+
4
0,0
(15)
L
x
5

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Compared with Equation 13 there exists hyperparameter vector 4p that controls the shift of the activation function. When * T, we cannot guarantee the conversion error is O. However, we can estimate the expectation of conversion error. Similar to (Deng & Gu 2020 we assume that of is uniformly distributed within intervals [( /T (1) and [ /L, (1) for
t= 1=,,,,... ..., 7 and I, = 1,2, I,, we have the following Theorem.
Theorem 2. An ANN with activation function [15 is converted to an SNN with the same weights. If 01 X, o' (0) d',, then for arbitrary T and I., the expectation of conversion error reaches when the shift term up in source ANN is 2'
VT, L
E2
Err
1== O.
(16)
The proof is in the Appendix. Theorem 2 indicates that the shift term 1 is able to optimize the expectation of conversion error. By comparing Figure 2 and Tigure[2. we can find that when the shift term + = 0.5 is added, the mean conversion error reaches zero, even though L # T. These results indicate we can achieve perforrefrrennce converted SNN ultra-low time-steps.
2
L is the only undetermined hyperparameter of the quantization clip-floor-shift activation. When T L. the conversion error reaches zero. So we naturally think that the parameter should be set as small as possible to get better performance at low time-steps. However, a too low quantization of the activation function will decrease the model capacity and further lead to accuracy loss when the time-steps is relatively large. Choosing the proper I. is a trade-off between the accuracy at low latency and the best accuracy of SNNs. We will further analyze the effects of quantization steps in the experiment section.
4.3 ALGORITHM FOR TRAINING QUANTIZATION CLIP-HLOOR- ACTIVATION FUNCTION
Training an ANN with quantization clip-floor-hhift activation instead of ReLU is also a tough prob- lem. To direct train the ANN, derivative of the floor function, that is dizl
the straight-through estimator (Bengio et al. 2013) for the
1. The overall derivation rule is given in Equation 17
de
The [1. 1, if x
2I <
<
N
ohu(=') Ox
21 if (arja otherwisc
< <
X
21
2L
02! 0, otherwise
(17)
Here 2 is the i-th element of 2!. Then we can train the ANN with quantization clip-floor-shie activation using Stochastic Gradient Descent algorithm (Bottou 2012).
5 RELATED WORK
The study of ANN-SNN conversion is first launched by Cao et al. (2155). Then Diehl et al. (2015) converted a lhyee-layer CNN to an SNN using data-based and model-based normalization. To ob- tain tigh-perforrmacee SNNs for complex datasets and deeper networks, Rueckauer et al. (016)
and [Sengupta et al. (2019) proposed more accurate scaling methods to normalize weights and scale thresholds respectively. which were later proved to be equivalent (Ding et al.) 2021). Nevertheless, the converted deep SNN requires hundreds of time steps to get accurate results due to the conver- sion error analyzed in Sec. 3 To address the potential information loss, Rueckauer al. (2016) neurons rather than "reset-to-zero" neurons. Recently, many methods have been proposed to eliminate the conversion error. Rueck- auer et al. (2016) recommended 99.9% percentile of activations as scale factors, and Ho & Chang (2020) added the trainable clipping layer. Besides, Han al. (2020) rescaled the SNN thresholds to avoid the improper activation of spiking neurons. [Masaa et al. (2020) and (Singh et al. (2021) evaluated the performance of converted SNNs on the Loihi Neuromorphic Processor. Our work share similarity with Deng & Gui (200)) Li et al. (202T). which also shed light on the conversion error. Deng & Gu 2020) minimized the wise error by introducing extra bias in addition to the converted SNN biases. Li al. I202I) further proposed calibration for weights and biases using quantized minettning. They got good results with 16 and 32 time-steps without trails for more ex- treme time-steps. In comparison, our work aims fit ANN into SNN with techniques eliminating
and Han et al.] (2020) suggested using greset-by-subtraction

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VGG-15 CIFAR-IO
CIFAR-1O
VGG-15 CIFAR-100
ReaNok20 CIFAR-100
Figure 3: Compare ANNs accuracy.
the mentioned conversion error. The end-to-end training of quantization layers is implemented to get better overall performance. Our shift correction can lead to a single SNN which performs well at both ultra-low and large time-steps. Maintaining SNN performance within extremely few time-steps is difficult even for supervised learning methods like backpropagation through time (BPTT). BPTT
usually requires fewer time-steps because of thorough training. yet at the cost of heavy GPU compu-
tation (Wu et al. 2018 2019; Lee et al. 20166] Neftci al., 2019: Lee et al., 2020 Zenke & Vogels,
2021 The timing- based backpropagation methods (Bohte et al. 2002 Tavanaei et al. 2019 Kim al. 2020) could train SNNs over very short temporal window, e.g. over 5- time-steps. How- ever, they are usually limited to simple datasets like MNIST (Kheradpisheh & Masquelier (2020)
and CIFARIO (Zhang Li, 2020 Rathi et al. (2019) shortened simulation steps by initializing SNN with conversion method and then tuning SNN with STDP. In this paper, the proposed method achieves perforreroommncc SNNs with ultra-low latency (4 time-steps).
6 EXPERIMENTS
In this section, validate the effectiveness of method and compare method with other state-of-the- approaches for image classification tasks on CIFAR- 10 (LeCun et al.) 1998). CIFAR- 100 (Krizhevsky et al. 2009), and ImageNet datasets (Deng et al. 2009). Similar to previous works, we utilize VGG-16 (Simonyan & Zisserman 2014). ResNet- 18 (He et al.] 2016). and ResNet-20 network structures for source ANNs. We compare our method with the state-of-the-art ANN-SNN conversion methods, including Hybrid- Sobbii-Conversion (HC) from Rathi et al. (2019), RMP from Han al. (2020). TSC from Han & Roy (2020), RNL from Ding et al. (2021). ReL.UhhreshlldShit (RTS) from Deng & Gu (020). and SNN Conversion with Advanced Pipeline (SNNC-AP) from Li ct al. (2021). Comparison with different SNN training methods also included to manifest the su- periority of low latency inference, including HybriddConversion-SSTDD (HC-STDB) from Rathi et al. (2019), STBP from Wu et al. (2018), DirectTraining (DT) from Wu et al. (2019), and TSSL from Zhang & Li 2020). The details of the proposed ANN-SNN algorithm and training configurations are provided in the Appendix.
6.1 TEST ACCURACY OF ANN WITH QUANTIZATION CLIP-FLOOR-SHIPE ACTIVATION
We first compare the performance of ANNs with quantization clip-floor activation (green curve), ANNs with quantization hiip-floor-shitie activation (blue curve), and original ANNs with ReLU acti- vation (black dotted line). Figure3] ad)- report the results about VGG-16 on CIFAR-10, 10, ResNet-20 on CIFAR-1O. VGG-16 on CIFAR- and ResNet-20 on CIFAR- 100. The performance of ANNS with quantization sipff activation is better than ANNs with quantization clip-floor activa- tion. These two ANNs can achieve the same performance as original ANNs with ReLU activation when L 4. These results demonstrate that our quantization hiip-flor--shf activation function hardly affects the performance of ANN.
6.2 COMPARISON WITH THE STATE- THE
Table 2 compares our method with the state-of-the-art ANN-SNN conversion methods on CIFAR- 10. As for low latency inference (T S 64), our model outperforms all the other methods with the same time-step setting. For T = 32, the accuracy of our method is slightly better than that of ANN (95.54% vs. 95.52%), whereas RMP, RTS, RNL, and SNNC-AP methods have accuracy loss of 33.3%, 19.48%, 7.42%, and 2.01%. Moreover, time-steps, which is 8 times faster than SNNC-AP that takes 32 time-steps. For ResNet-20, we achieve accuracy of 83.75% with 4 time-steps. Notably, ultra-low latency performance is comparable with other state-of-the of- supervised training methods, which is shown in Table [S3 of the Appendix.
achieve accuracy of 93, 93.96% using only 4
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Sendbien
Smnkkaniitnekkk (6) VGG-16 on CIEAR-100
(4) VGG-16 on CIEAR-10
(5) ReaNet-20 on CIEAR-10
(4) ReaNet-20 CIEAR-100
Figure 4: Compare quantization clip-floor activation with/without shift term
Table 2: Comparison between the proposed method and previous works on CIFAR-10 dataset.
Architecture
Method
ANN
T=2
T=
T=8
T=16
T=32
T=64
T2512
RMP
93.63%
60.30%
90.35%
93.63%
TSC 93.63%
92.79%
93.63%
RTS
95.72%
76.24%
90.64%
95.73%
VGG-16
RNL
92.82%
57.90%
85.40%
91.15%
92.95%
SNNC-AP
95.72%
93.71%
95.14%
95.79%
Ours
95.52%
91.18%
93.96%
94.95%
95.40%
95.54%
95.55%
95.59%
RMP
91.47%
91.36%
Re
TSC
91.47%
69.38%
91.42%
Ours
91.77%
73.20%
83.75%
89.55%
91.62%
92.24%
92.35%
92.41%
RTS
95.46%
84.06%
92.48%
94,42%
R8
SNNC-AP
95.46%
94.78%
95.30%
95.45%
Ours 96.04%
75.44%
90.43%
94.82%
95.92%
96.08%
96.06%
96.06%
SNNC-AP use ResNet- 18, ours use standard ResNet-118.
We further test the performance our method on the large-scale dataset. 3 the results on ImageNet, our method also outperforms the others both in terms of high accuracy and ultra-low latency. For ResNet-34, the accuracy of the proposed method is 4.83% higher than SNNC-AP and 69.28% higher than RTS when T = 32. When the time-steps is 16. of 59.35%. %. For VGG-16, the accuracy of the proposed method is 4.83% higher than SNNC-AP still achieve an
still achieve accuracy
and 68.356% higher than RTS when T 32. When the time-steps is 16, accuracy of 50.97%. These results demonstrate that our method outperforms the previous conversion methods. More experimental results CIFAR-100) is in Table S4 of the Appendix.
6.3 COMPARISON OF QUANTIZATION CLIP-FLOOR AND QUANTIZATION CLIP-FLLOOR-SHE
Here we further compare the performance of SNNs converted from ANNs with quantization clip- floor activation and ANN with quantization clop-floo--shift activation. In Sec. 4 we prove that the expectation of the conversion error reaches with quantization clip-floor-shif activation, no matter whether T and L are the same or not. To verify these, we set L to 4 and train ANNs with quantization clip-floor activation and quantization clip-floor-hifft activation, respectively. Figure 4 shows how the accuracy of converted SNNs changes with respect to the time-steps T. The accuracy of the converted SNN (green curve) from ANN with quantization clip-floor activation (green dotted line) first increases and then decreases rapidly with the increase of time-steps, because we cannot guarantee that the conversion error is zero when T not equal to L. The best performance still lower than source ANN (green dotted line). In contrast, the accuracy of the converted SNN from ANN with quantization clip- floffo-r-sshftt activation (blue curve) increases with the increase of T. It gets the same accuracy as source ANN (blue dotted line) when the time-steps is larger than 16.
6.4 EFFECT OF QUANTIZATION STEPS L
In our method, the quantization steps is a hyperparameter, which affects the accuracy of the con- verted SNN. To analyze the effect of L and better determine the optimal value, we train VGG- 16/ResNet-20 networks with quantization clip-floor-shift activation using different quantization steps L, including 2,4,8,16 and 32, and then converted them to SNNs. The experimental results on CIFAR- 10/100 dataset are shown in Table [22 and Figure where the black dotted line denotes the ANN accuracy and the colored curves represent the accuracy of the converted SNN. In order to
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Autemay &
Senabiie lin--kze (5) ReaNet-20 on CIEAR-10
Sandkariitnenkke (6) VGG-16 on CIFAR-100
(4) VGG-16 on CIEAR-10
(4) ReiNet-20 CIEAR-100
Figure 5: Influence of different quantization steps
Table Comparison between the proposed method and previous works on ImageNet dataset.
Architecture
Method
ANN
T=16
T=32
T=64
T=128
T=256
T2 1024
RMP
70.64%
65.47%
TSC
70.64%
61.48%
65.10%
ResNet-34
RTS
75.66%
0.09%
0.12%
3.19%
47.1
75.08%
SNNC-AP
75.66%
64.54%
71.12%
73.45%
74.61% %
75.45%
Ours
74.32%
59.35%
69.37%
72.35%
73.15%
73.37%
73.39%
RMP
73.49%
48.32%
73.09%
TSC
73.49%
69.71%
73.46%
VGG-16
RTS
75.36%
0.114%
0.118%
0.122%
1.81%
73.88%
SNNC-AP
75.36%
63.64%
70.69%
73.32%
74.23%
75.32%
Ours
74.29%
50.97%
68.47%
72.85%
73.97%
74.22%
74.32%
balance the trade-off between low latency and high accuracy, we evaluate the performance of con- verted SNN mainly in two aspects. First, we focus on the SNN accuracy at ultra-low latency (within 4 time-steps). Second, we consider the best accuracy of SNN. It is obvious to find that the SNN accuracy at ultra-low latency decreases as increases. However, a too small will decrease the model capacity and further lead to accuracy loss. When L 2, there exists clear gap between the best accuracy of SNN and source ANN, The best accuracy of SNN approaches source ANN when L 4. In conclusion, the setting of parameter mainly depends on the aims for low latency best accuracy. The recommend quantization step L is or 8, which leads to perforerfor converted SNN at both small time-steps and very large time-steps.
DISCUSSION AND CONCLUSION
In this paper, present ANN-SNN conversion method, enabling high-accuracy and ultra-low- latency deep SNNs. We propose the quantization clip-floor-shift activation to replace ReI.U activa- tion, which hardly affects the performance of ANNs and is closer SNNs activation. Furthermore, we prove that the expected conversion error is zero, no matter whether the time-steps of SNNs and the quantization steps of ANNs is the same not. We achieve state-of-the-art accuracy with fewer time-steps on CIFAR- 10, CIFAR-100 100, and ImageNet datasets. Our results can benefit the imple- mentations on neuromorphic hardware and pave the way for the large-scale application of SNNs.
Different from the work of Deng & Gu 2020). which adds the bias of the converted SNNs to shift the theoretical ANN-SNN curve to minimize the quantization error, we add the shift term in the quantization clip-floor activation function, and use this quantization clip-floor-hhift function to train the source ANN. We show that the shift term can overcome the performance degradation problem when the time-steps and the quantization steps not matched. Due to the unevenness error, there still exists a gap between ANN accuracy and SNN accuracy, even when L = T. Moreover, it is hard to achieve gigghpeeffrmance ANN- -NN conversion when the time-steps T 1. All these problems deserve further research. One advantage of conversion -aased methods is that they can reduce the overall computing while maintaining comparable performance as conversion- based methods and model compression may help significantly reduce the neuron activity and thus reduce energy consumptions without suffering from accuracy loss (Kundu al. 2021 Rathi & Roy 2021), which is a promising direction.
ANN. Combining the

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ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China under contracts No.62176003 and 12088208....
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A APPENDIX
A.1 NETWORK STRUCTURE AND TRAINING CONFIGURATIONS
Before training ANNs, we first replace max-pooling with average- pooling and then replace the ReLU activation with the proposed quantization clip-floor-shift activation (Equation [5]. After training, copy all weights from the source ANN the converted SNN, and the threshold of in each layer of the converted SNN equal to the maximum activation value X of the source ANN in the same layer. Besides, we set the initial membrane potential v' (0) in converted SNN as 0 /2/2to to match the optimal shift & 2 of quantization clip-floor-shift activation in the source ANN.
Despite the common data normalization, we use some data pre-processing techniques. For CIFAR datasets, we resize the images into 32 x 32, and for ImageNet dataset, we resize the image into 224 224. Besides, we use random crop images, Cutout (Devries & Taylor 2017) and AutoAug- ment (Cubuk et al. 2019) for all datasets.
We use the Stochastic Gradient Descent optimizer (Bottou 2012) with momentum parameter of 0.9. The initial learning rate is set to 0.1 for CIFAR-10 and ImageNet, and 0.02 for CIFAR-100. A cosine decay scheduler (Loshchilov & Hutter, 2016) is used to adjust the learning rate. We apply a 5 x 10-4 weight decay for CIFAR datasets while applying 1 x 10 weight decay for ImageNet. We train all models for 300 epochs. The quantization steps is set to when training the networks CIFAR-10, and VGG-16, 16, ResNet-18 CIFAR- 100 dataset. When training ResNet-20 on CIFAR- 100, the parameter L is set to 8. When training ResNet-34 and VGG-16 on ImageNet, the parameter I. is set to 8, 16, respectively. We constant input when evaluating the converted
SNNs.
A.2 INTRODUCTION OF DATASETS
CIFAR-10. The CIFAR-10 dataset (Krizhessky et al. 10 classes. There are 50000 training images and 10000 test images.
2009) consists of 60000 32 x 32 images in
CIFAR-100. The CIFAR-100 dataset (Krizhessky et
2009 consists of 60000 32 x 32 images in
100 classes. There are 50000 training images and 10000 test images.
ImageNet. We use the ILSVRC 2012 dataset (Russakovsky et al. 2015), which consists 1,281,167 training images and 50000 testing images.
A.3 DERIVATION OF EQUATION 12 AND PROOF OF THEOREM 2
Derivation of Equation 1
Similar to x' W'aa' We define
n 2'(1) = W ('))-(())
(S1)
We use (t) and of to denote the i-th element in vector ut(t) and z', respectively. To derive Equation M some extra assumptions on the relationship between ANN activation value and SNN postsynaptic potentials are needed, which are showed in Equation S2
{
if 2 < 0, then Vi (1) < 0, if & 2f < 0j, then Vt 0 < (t(()) & 0
(S2)
if 2f > 0,, then Vi (1( > 0.
With the assumption above, we can discuss the firing behavior of the neurons in each time-step. When af < 0 or 2/ > 0, the neuron will never fire fire all the time-steps, which means (T( or (T) = 01. In this situation, we can use clip function to denote (!()).
(T) clip(2/, 0,0")
(S3)
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When 0 < 2f < B%, every imput from the presynaptic neuron in SNNs falls into [0, 6'], then have VI, (1) E [0, 0). We can rewrite Equation into the following equation.
(TYT 0
z + (0)
(T)
(S4)
0
g
Considering that
STTT 0x
Ex
s1(1)=
and 0 < x((T) 0x
Equation is changed to:
0²
T (0)
(T
(S5)
We combine these two situations (Equation S3
and Equation S4).
and we have:
z'T + r (0)
('(T) 0¹ clip
.0.1)
(S6)
T
A
Proof of Theorem
Before prove Theorem 2 we first introduce Lemma
Lemma 1. If random variable E [0, 0] is uniformly distributed in every small interval [m,,, M
with the probability density function (t = 0, 1..,,..T T), where my = 0, m1++1 = 0, = (t-
for 1,2,... .... T, pa Pr, we can conclude that
0
Tx
1
E5
+
0.
(S7)
T
2
Proof.
0
Tx
1
0/27
0
xT
1
E
T
+
2
Po
T
2)
dx
T
12/+1)
0
IT
1
(21 (21-1)01 1)0/2T
Dt
T
2
dx
0
xT
1
PT
T
0
2
da
(TT 17T-17 1/0/2T
8/2T
T-1
(20+1)0/27
1
Po
T. dx +
Σ Ph
(x
dx + pr
(x - 0) dr
(21 (21-1)0/21
T
120-1)0/22
0
Pa
PT
(po Pr)
87 O.
(S8)
872
87
Theorem 2. An ANN with activation function (15 is converted SNN with the same weights If X 00) = 0 sp, then for arbitrary T and L, the expectation of conversion error reaches when the shift term yp in source ANN is 2'
VT,L L E, Err
= O.
(S9)
1p=
Proof.
10'
zT
4 0)=
E,
Err
E.
(S10)
-=
g
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Potenial
Time-stop
Figure S1: More spikes than expected exists for the method of setting the maximum activation.
As every element in vector z is identical, we only need to consider one element.
0c
T
(0)
X'
z sp
Eg
+
T
0
L
x
gl
(0)
X L +
ppi 1)))
= Es,
2
+ Ez,
2
(S11)
T
0
L
x
According to Lemma
we have
0
T
(0)
E2
-2!
0,
(S12)
T
11/2
L
1
E
Pi
(S13)
L
4=1/2
Thus the sum of both terms also equals zero
A.4 COMPARISON OF THE METHODS WITH OR WITHOUT DYNAMIC THRESHOLD ON THE CIFAR-100 DATASET
In this paper we use training parameter X to decide the maximum value of ANN activation. The previous works suggested to set the maximum value of ANN activation after training as the threshold. If we set 0' = (0.1" (max(('-1 (anx)))''ss)), the situation of fewer spikes expected never happens, as we can prove that (TT) 0' (see Theorem 3). Despite this, there still exists the situation of more spikes as expected. An example is given in Figure [ST Here we consider the same example in Figure [ In source ANN, we suppose that two analog neurons layer I 1-1 1 are connected to an analog neuron in layer with weights 2 and -2, and the output vector a of neurons in layer - 1 is 00.6, 0.4). Besides, in converted SNN, we suppose that the two spiking neurons in layer 1 1 fire 3 spikes and 2 spikes in 5 time-steps (T=5), respectively, and the threshold g!-1 = 1. Thus, (
Σr s' (1) gl- [0.6, 0.4]. According to Equation | 0.4. As for SNN, we suppose that the presynaptic neurons fires at t = 1, 3 and 4, 5, respectively. Even through we set the threshold
T
the ANN output a' '' = [2 1..66.0600.4
g = 1 to the maximum activation 2, the postsynaptic neuron will fire three spikes at t 1,2, 3, and
o' (T) = 0.6 > a'.
Besides, setting max (max(0I-IW's)) as the threshold brings two other problems. First, the spiking neurons will take a long time to fire spikes because of the large value of the threshold, which makes it hard to maintain SNN performance within a few time-steps. Second, the quantization emoor will be large it is proportional to the threshold. If the conversion error is not zero for one layer, it will propagate layer by layer and will be magnified by larger quantization errors. We compare method and the method of setting the maximum activation the CIFAR- 100 dataset. The results are reported in Table where DT represents the dynamic threshold in our method. The results show that our method can achieve better performance.
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Table S1: Comparison between our method and the method of setting the maximum activation.
DT'
w/o shift
T=4
T=8
T=16
T=32
T=64
T=128
T=256
T>512
VGG-16 16 CIFAR- 1FAAR-100 with L=4
69.62%
73.96%
76.24%
77.01%
77.10%
77.05%
77.08%
77.08%
21.57%
41.13%
58.92%
65.38%
64.19%
58.60%
52.99%
49.41
x
1.00%
0.96%
1.00%
1.10%
2.41%
13.76%
51.70%
77.10%
X
1.00%
1.00%
0.90%
1.00%
1.01%
2.01%
19.59%
70.86%
Dynamic threshold.
Theorem 3. If the threshold is set the maximum value of ANN activation, that is O max4c(0.1)) (max(6'-1W''s and (0 < l:. Then at any time-step, the membrane potential of each neuron after spike (t) will be less than 6', where represents the index of each neuron.
Proof. We prove by induction. For O, it is easy to see (0 6'. For O, we suppose that u/ (((t-1) 1) < H' Since we have set the threshold to the maximum possible input, and (1) represents the imput from layer I to the i-th neuron in layer I, x (t) will be no larger than # for arbitrary t. Thus we have
(t( df(t - 1) of <00 0 + = =20'
(S14)
(1) (1m((())0'),
(S15)
(1) = (((1)-$(110)
(S16)
I < (t(() < 20', then we have (t = (t) 00 < 8¹. If( < 0,, then u((() = m!(t) < 0
By mathematical induction, (1) 8 holds for any >
A.5 EFFECT OF QUANTIZATION STEPS I.
Table S2 reports the performance of converted SNNs with different quantization steps L and differ- ent time-steps T. For VGG-16 and quantization steps L 2, we achieve an accuracy of 86.53% on CIFAR-10 dataset and an accuracy of 61.4 61.41% on CIFAR-100 dataset with time-steps. When the quantization steps = 1, we cannot train the source ANN.
A.6 COMPARISON WITH STATF-OF-THR-ARY SUPERVISED TRAINING METHODS ON CIFAR-10 DATASET
Notably, our ultra-low latency performance is comparable with other state-of-the-arr supervised training methods. Table [S3 reports the results of hybrid training and backpropagation methods on CIFAR-10. The backpropagation methods require sufficient time-steps to convey discriminate information. Thus, the list methods need at least 5 time-steps to achieve ~91% accuracy. On the contrary, our method method requires 200 time-steps to obtain 92.02% accuracy because of further training with STDB, whereas our method achieves 93.96% accuracy with 4 time-steps.
achieve 94.73% accuracy with 4 time-steps. Besides, the hybrid training
A.7 COMPARISON ON CIFAR-100 DATASET
Table S4 reports the results CIFAR-100, our method also outperforms the others both in terms of high accuracy and ultra-low latency. For VGG-16, the accuracy of the proposed method is 3.46% higher than SNNC-AP and 69.37% higher than RTS when T 32. When the sime-steps is only 4, we can still achieve an accuracy of 69.62%. These results demonstrate that our method outperforms the previous conversion methods.
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Table S2: Influence of different quantization steps.
quantization steps
T=1
T=2
T=4
T=8
T=16
T=32
T=64
T=128
VGG-16 on CIFAR-10
L=2
86.53%
91.98%
93.00%
93.95%
94.18%
94.22%
94.18%
94.14%
L=4
88.41 %
91.18%
93.96%
94.95%
95.40%
95.54%
95.55%
95.59%
L=8
62.89%
83.93%
91.77%
94.45%
95.22%
95.56%
74.%
95.79%
L=16
61.48%
76.76%
89.61%
93.03%
93.95%
94.24%
94.25%
94.22%
L=32
13.05%
73.33%
89.67%
94.13%
95.31%
95.66%
95.73%
95.77%
ResNet-20 on CIFAR-10
L=2
77.54%
82.
85.77%
88.04%
88.64%
88.79%
88.85%
88.76%
L==4
62.43%
73.2%
83.75%
89.55%
91.62%
92.24%
92.35%
92.35%
L=8
46.19%
58.67%
75.70%
87.79%
92.14%
93.04%
93.34%
93.24%
1.=16
30.96%
39.87%
57.04%
79.5%
90.87%
93.25%
93.44%
93.48%
I=32
22.15%
27.83%
43.56%
70.15%
88.81%
92.97%
93.48%
93.48%
VGG-16 on CIFAR-II R-000
L=2
61.41% %
64.96%
68.0%
70.72%
71.87%
72.28%
72.35%
72.4%
L=4
57.5%
63.79%
69.62%
73.96%
76.24%
77.01%
77.1%
77.05%
L=8
44.98%
52.46%
62.09%
70.71%
74.83%
76.41%
76.73%
76.73%
L=16
33.12%
41.71%
53.38%
65.76%
72.80%
75.6%
76.37%
76.36%
L=32
15.18%
21.41%
32.21%
50.46%
67.32%
74.6%
76.18%
76.24%
ResNet-20 on CIFAR-100
L=2
38.65%
47.35%
55.23%
59.69%
61.29%
61.5%
61.03%
60.8 %
L=4
25.62%
36.33%
51.55%
63.14%
66.70%
67.47%
67.47%
67.41% %
I.=8
13.19%
19.96%
34.14%
55.37%
67.33%
69.82%
70.49%
70.55%
L=16
6.09%
9.25%
17.48%
38.22%
60.92%
68.70%
70.15%
70.20%
L=32
5.44% %
7.41%
13.36%
31.66%
58.68%
68.12%
10%
70.27%
A.8
ENERGY CONSUMPTION ANALYSIS
We evaluate the energy consumption of our method and the compared methods (Li et al. 2021 Deng & Gu 2020) CIFAR-100 datasets. Here
the same network structure of VGG- 16. Following the analysis in Merolla et al. (2014), we use synaptic operation (SOP) for SNN to represent the required basic operation numbers classify one image. We utilize 77EJ/SOP for SNN and 12.5pI/FLOP for ANN as the power consumption baseline, which is reported from the ROLLS neuromorphic processor (Qiao al.) 2015). Note that we do not consider the memory access energy in our study because it depends the hardware. As shown in Table (S5] when the time-steps is the same, the energy consumption of our method about two times of SNNC-AP, However, to achieve the same accuracy of 73.55% our method requires less energy consumption.
A.9 PSEUDO-CODE FOR OVERALL CONVERSION ALGORITHM
In this section, we summarize the entire conversion process in Algorithm including training ANNs from scratch and converting ANNs SNNs. The QCFS in the pseudo-code represents the proposed quantization fli-fllor-shift function.
17

---

Published as a conference paper at ICLR 2022
Table S3: Compare with state-of-the-art supervised training methods on CIFAR-10 dataset
Model
Method
Architecture
SNN Accuracy
Timesteps
CIFAR-10
HC
Hybrid
VGG-16
92.02
200
STВP
Backprop
CIFARNet
90.53
12
DT
Backprop
CIFARNet
90.98
8
TSSL
Backprop
CIFARNet
91.41
5
DThIR!
ANN-SNN
cNet
77.10
256
Ours
ANN- SNN
VGG-16
93.96
Ours
ANN-SNN
CIFARNet
94.73
Implemented on Loihi neuromorphic processor For CIFARNet, we use the same architecture as Wu et al.
2018
Table S4: Comparison between the proposed method and previous works on CIFAR-100 dataset.
Architecture
Method
ANN
T=2
T=4
T=8
T=16
T=32
T=64
T2512
RMP
71.22%
70.93%
TSC
71.22%
70.97%
VGG-16
RTS
77.89%
7.64%
21.84%
77.71%
SNNC-AP
77.89%
73.55%
76.64%
77.87%
Ours 76.28%
63.79%
69.62%
73.96%
76.24%
77.01%
70.10%
77.08%
RMP
68.72%
27.64%
46.91 %
67.82%
ResNet-20
TSC
68.72%
68.18%
Ours
69.94%
19, 99.96%
34.14%
55.37%
67.33%
69.82%
70.49%
70.50%
RTS
77.16%
51.27%
70.
77.19%
R8sNet-18
SNNC-AP
77.16%
76.32%
77.29%
77.25%
Ours
78.80%
70. 70.79%
75.67%
78.48%
79.48%
79.62%
79.54%
79.61%
RTS and SNNC-AP altered ResNet- 18, while ours standard ResNet-18.
Table S5: Comparison of the energy consumption with previous works
Method
ANN
T=2
T=4
T=8
T=16
T=32
T=64
Accuracy
77.89%
7.64%
21.84%
RTS
OP (GFLOP/GSOP)
0.628
0,508
0.681
Energy (mJ)
7.85
0.039
0.052
Accuracy
77.89%
73.55%
76.64%
SNNC-AP
OP (GFLOP/GSOP)
0.628
0.857
1.22
Energy (mJ)
7.85
0.660
0.094
Accuracy
76.28%
63.79%
69.62%
73.96%
76.24%
77.01% %
77.10%
Ours
OP (GFLOP/GSOP)
0.628
0.094
0.185
0.364
0.724
1.444
2.884
Energy (mJ)
7.85
0.007
0.014
0.028
0.056
0.111
0.222
18

---

Published as a conference paper at ICLR 2022
Algorithm Algorithm for ANN-SNN conversion.
Input: ANN model (a: W) with initial weight W; Dataset D; Quantization step L; Initial dynamic thresholds A; Learning rate
Output: MsNN (a:; W)
1: for = 1 to MaNe.layers do
2: if is ReLU activation then
3: Replace ReI.U(x)) by QCFS(a;; A")
4: end if
5: is MaxPooling layer then
6: Replace MaxPooling layer by AvgPooling layer
7: end if
8: end for
9: for = to epochs do
for length of Dataset D do
11:
Sample minibatch (æ:" y) from D 12: for to MANN do
13:
x' QCCS(WWLS end for
X
14:
15:
Loss CrossEntropy((r* a. y)
16:
for = to MANN layers do
W + W 2Loss
OW
18: N < X e OX
aLoss
19:
end for
20: end for
21: end for
22: for 1. to layers do
23: MSNN. W + MANN. W
MsNN. +- MANN-X
25: MSNN. V (0) + MsNN. /2
end for
27:
return MsNN
19


Reviews:
[{"review_id": "vl36MFdfiau", "reviewer": "Reviewer_LAGa", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "Following are my detailed reviews.\n\nStrenghts\n========\n1. The paper is well written and the general motivation of work on efficient ANN-SNN conversion is important as it can reduce the SNN training epoch over head. In particular, the general trend is that only ANN-SNN conversion asking for higher times steps (latency). So, achieving reduced latency for only conversion based approach is a promising direction.\n\n2. The organization of the paper and results are good.\n\nWeaknesses:\n===========\n1. The variable $\\textbf{v}^l(t)$ is defined twice and are not having an exactly similar definition. Please clarify whether its membrane potential at time $t$ or membrane potential at $t$ after spike output happens. The definition of $\\textbf{m}^l(t)$ is not clear either, what is meant by membrane potential after neuronal dynamics? I think, the latter is before applying the threshold ${\\theta}^l$ and the earlier is after applying that as per Eq. 5. So, please clarify this part. Also, if you are not using the return to zero mechanisms, why is Eq. 5 valid?  Shouldn't it have ${\\theta}^l_{reset}$? Or else clearly mention that $\\theta^l$ is $\\theta^l_{th}$ - ${\\theta}^l_{reset}$.\n\n2. An important way to reduce spiking activity is through weight sparsity. It would be interesting to see whether such \"only conversion\" based strategy go hand in hand with state-of-the-art model compression of SNNs [1, 2]. As, compression is a necessary part for model size reduction and is equally important as considering reduction of latency for edge deployment for real time applications.\n\n3. I am not 100% sure about the way unevenness error has been portrayed here. I think its largely a function of what the max. weight value and the threshold is. In general we set the max. $W \\times x$ value as the threshold. Where as in the example the authors has taken 1 as threshold and 1.5 as weight. Please also provide analysis with threshold as max ( $W \\times x$). Also, I think the goal of trainable thresholds [2] should be to mitigate such error by default when SNN training happens. The authors should provide at least some discussion on that to clearly justify its inception and other ways of possible mitigation. Also, when the pre-synaptic spike is 1 at t=1, then the potential starts accumulating, then why it should immediately reflect at output at t = 1? If the inputs spikes at t = 1 then shouldn't the output postsynaptic spike happen at t = 2? I am not sure whether the Fig, 1(b) and (c) is correct. I think presynaptic red pulses should be at 0 and 2. If that's not the case, please justify why.\n\n4. As the authors claimed the activation difference error is zero if the ANN is trained with the proposed activation non-linearity, then is it possible to convert with  T = 1? Please show results for T= 1 as that would make the limitations and possibilities much clearer. \n\n5. Gu et al. [3] has also suggested ReLU shifting function in their optimal conversion work, how does the Therorem 2's shifting proposal differ from theirs, please highlight that.\n\n6. One part may be confusing to the reader is that we generally evaluate threshold after ANN training, i that case how can we perform the proposed non-linear activation efficiently during ANN training. Please clearly state tis in the paper.\n\n7. Please mention clearly how many training or test images you used for conversion. \n\n[1]  Spike-thrift: Towards energy-efficient deep spiking neural networks by limiting spiking activity via attention-guided compression, WACV 2021.\n[2] DIET-SNN: A LOW-LATENCY SPIKING NEURAL NETWORK WITH DIRECT INPUT ENCODING & LEAKAGE AND THRESHOLD OPTIMIZATION, IEEE TNNLS 2021.\n[3] Optimal conversion of conventional artificial neural networks to spiking neural networks, ICLR 2021.\n", "summary_of_the_paper": "The paper first analyzed the various sources of ANN-SNN conversion errors namely clipping error, quantization error and unevenness error and then proposed quantization clip-floor-shift activation function to replace the ReLU activation function in source ANNs to train them. This resulted in ANN-SNN conversion at ultra low latency (for example t step of 4).", "summary_of_the_review": "The paper's effort to provide more similar ANN activation as SNN to reduce conversion error of ANN-SNN at low time step is interesting and promising. However, there are some concerns as I have detailed in my review that can further clarify the paper for good.", "strength_and_weaknesses": "", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "gsUd019Lfr_", "reviewer": "Reviewer_hn22", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "1.\tThere are other related works that need to be compared with this method. For example:\n\na.\tR. Massa, A. Marchisio, M. Martina and M. Shafique, \"An Efficient Spiking Neural Network for Recognizing Gestures with a DVS Camera on the Loihi Neuromorphic Processor,\" 2020 International Joint Conference on Neural Networks (IJCNN), 2020, pp. 1-9, doi: 10.1109/IJCNN48605.2020.9207109.\n\nb.\tS. Singh, A. Sarma, S. Lu, A. Sengupta, V. Narayanan and C. R. Das, \"Gesture-SNN: Co-optimizing accuracy, latency and energy of SNNs for neuromorphic vision sensors,\" 2021 IEEE/ACM International Symposium on Low Power Electronics and Design (ISLPED), 2021, pp. 1-6, doi: 10.1109/ISLPED52811.2021.9502506.\n\n2.\tFigure 3 is not clear. It seems that the legend is missing to explain what is represented by the lines.\n3.\tSince they represent an important contribution, the Imagenet results should be reported and discussed in main manuscript, instead of placing them in the supplementary material.\n4.\tIn Section 6, it can be observed that the ResNet-20 is the only considered CNN model in which the accuracy with T=4 is significantly lower than the accuracy with larger T. The reason should be discussed more comprehensively in the supporting text.\n", "summary_of_the_paper": "This research proposes an ANN-SNN conversion method based on quantization clip-floor-shift activation.", "summary_of_the_review": "The achieved experimental results are notable. However, some key points need to be addressed (see the main review section).", "strength_and_weaknesses": "", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "eG58v2Q2cQ1", "reviewer": "Reviewer_d5di", "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": "**Pros**:\n\n+ This paper gives a theoretical condition of zero conversion error. Under a uniform prior assumption, the authors propose to shift the activation and get zero expected error.\n\n+ The empirical results show good improvements. \n\n**Cons**:\n\n- This is a little suggestion: this paper has too many notations, it would be better to have a section or table to sort them out. \n\n- The analysis of conversion error has been proposed in Li et al. 2021. The only difference is the unevenness error where Li et al. 2021 assume it is 0. This paper is over-claiming and should point this out. \n\n- What is the difference between shift term $\\varphi$ in this paper and the optimal shift in Deng & Gu., 2020? Afaic, the uniform condition and the function of the shift are the same. \n\n- Lack of ImageNet conversion experiments. The authors are strongly encouraged to convert more challenging models on the ImageNet dataset. Therefore, I would reject this paper for now.\n\n", "summary_of_the_paper": "This paper proposes a Quantization neural network to spiking neural network (QNN2SNN) conversion method. The authors first analyze the conversion error between ANN and SNN. Then they construct the ann with quantized activation so that the error can be eliminated. Both theoretical and empirical results are presented in this paper. ", "summary_of_the_review": "My weakness is listed above. Besides, I also have some questions:\n\n1. If Threorm 1 & 2 are correct, why your model does not reach exactly the same accuracy as ANN when $L=T$?\n2. What is energy consumption? It is higher or lower?", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "ZB5UxgUBGNy", "reviewer": "Reviewer_5DgR", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "Strengths:\n(1) Excellently presented paper that provides a full explanation of the development in SNNs.\n(2) The idea of obtaining a high-performance SNN from a quantization ANN is intuitive. However, the converted SNN will suffer a performance degradation problem if the time-steps of SNNs dose not match up with the quantization step of ANNs. The authors propose to add a shift term to overcome this problem, which is very interesting and impressive.\n(3) The proposed framework is rigorous, and is supported by theoretical analysis.\n(4) The proposed approach achieves SOTA accuracy with fewer time-steps.\n \nWeaknesses & suggestions for improvement:\n(1) My main concern is about unevenness error. Is there such a case, the input is negative during a period as the weight is negative? Thus, the spiking neurons fire fewer spikes than we expected.\n(2) The accuracy of source ANNs on ImageNet is lower than some compared methods, which decreases the performance of converted SNNs.  \n(3) The accuracy of converted SNNs from ANNs with quantization clip-floor-shift activation (blue curve in Fig. 4c) is slightly higher than the source quantization ANN when the time-steps is large. I would like to know whether the performance is the same as or better than source ANN without quantization when the time-steps is large.\n(4) I would like to see the discussion of the limitation of ANN-SNN conversion.\n", "summary_of_the_paper": "This paper proposes a quantization clip-bottom-shift activation function to replace the ReLU activation function in ANNs, so as to better approximate the activation function of SNNs. The authors also prove that the expected error of ANN-SNN conversion can be reduced to 0 by using this method. The reported results on CIFAR-10/100 and ImageNet dataset show that this work archives state-of-the-art accuracy with fewer time-steps.", "summary_of_the_review": "This paper proposes a novel ANN-SNN conversion framework and achieves SOTA accuracy with fewer time-steps. The suggestion for improvement is listed above.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and 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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors present an improved method to convert ANNs to spiking neural networks (SNNs). First, a network with quantized activations is constructed, then it is converted. They analyze the conversion errors theoretically. In addition to previously considered errors [Li et al. 2021] they also consider an error they call "unevenness error" and propose a way to compensate for that.
They test the method on data sets such as CIFAR-100 and show good improvements over previous methods with respect to classification accuracy and inference time.
The reviewers agree that the manuscript presents interesting and valuable work with a significant novel contribution.The manuscript is well written.

Weak points according to the first reviews were:
- Lack of ImageNet conversion experiments.
- Analysis of energy consumption was missing.
- More related work needs to be compared.
The revision addressed all these points, This was acknowledged by the reviewers with increased ratings. All reviewers propose acceptance.
Decision: Accept (Poster)