Bridging the Gap between ANNs and SNNs by Calibrating Offset Spikes
Authors: Zecheng Hao
Jianhao Ding
Tong Bu
Tiejun Huang
Zhaofei Yu
Venue: ICLR 2023 poster (2023)
Keywords: Spiking Neural Networks，Spike Calibration，Ultra-low-latency Conversion

Abstract: Spiking Neural Networks (SNNs) have attracted great attention due to their distinctive characteristics of low power consumption and temporal information processing. ANN-SNN conversion, as the most commonly used training method for applying SNNs, can ensure that converted SNNs achieve comparable performance to ANNs on large-scale datasets. However, the performance degrades severely under low quantities of time-steps, which hampers the practical applications of SNNs to neuromorphic chips. 
In this paper, instead of evaluating different conversion errors and then eliminating these errors, we define an offset spike to measure the degree of deviation between actual and desired SNN firing rates. We perform a detailed analysis of offset spike and note that the firing of one additional (or one less) spike is the main cause of conversion errors. Based on this, we propose an optimization strategy based on shifting the initial membrane potential and we theoretically prove the corresponding optimal shifting distance for calibrating the spike. In addition, we also note that our method has a unique iterative property that enables further reduction of conversion errors. The experimental results show that our proposed method achieves state-of-the-art performance on CIFAR-10, CIFAR-100, and ImageNet datasets. For example, we reach a top-1 accuracy of 67.12% on ImageNet when using 6 time-steps. To the best of our knowledge, this is the first time an ANN-SNN conversion has been shown to simultaneously achieve high accuracy and ultralow latency on complex datasets. Code is available at https://github.com/hzc1208/ANN2SNN_COS.

Published as a conference paper at ICLR 2023
BRIDGING THE GAP BETWEEN ANNS AND SNNS BY CALIBRATING OFFSET SPIKES
Zecheng Hao', Jianhao Ding', Tong B Tiejun Huang!? 1. & Zhaofei Yu!
School of Computer Science, Peking University
2 Institute for Artificial Intelligence, Peking University
ABSTRACT
Spiking Neural Networks (SNNSs have attracted great attention due to their dis- tinctive characteristics of low power consumption and temporal information pro- cessing. ANN-SNN conversion, as the most commonly used training method for applying SNNs, can ensure that converted SNNs achieve comparable performance ANNs on large-scale datasets. However, the performance degrades severely under low quantities of time-steps, which hampers the practical applications of SNNs to neuromorphic chips. In this paper, instead of evaluating different conver- sion errors and then climinating these errors, we define an offset spike to measure the degree of deviation between actual and desired SNN firing rates, We perform a detailed analysis of offset spike and note that the firing of one additional (or one less) spike is the main cause of conversion errors. Based on this, we pro- pose an optimization strategy based on shifting the initial membrane potential and we theoretically prove the corresponding optimal shifting distance for calibrating the spike. In addition, we also note that our method has a unique iterative prop- erty that enables further reduction of conversion errors. The experimental results show that our proposed method achieves state-of-the-a performance on CIFAR- 10. CIFAR- 100, and ImageNet datasets. For example, we reach a top-1 accuracy of 67. 12% on ImageNet when using 6 time-steps. To the best of our knowledge, this is the first time an ANN-SNN conversion has been shown to simultaneously achieve high accuracy and ultralow latency on complex datasets. Code available at wttpt::t//// /ii hub. cww www.o///ccc00... wttp://22220/AA2NN___O..
INTRODUCTION
Acclaimed as the third generation of Artificial Neural Networks (Maass, 1997), Spiking Neural Networks (SNNs) have brought brand-new inspiration to computational neuroscience. As the corre- sponding neuron fires spikes only when the current membrane potential exceeds the firing threshold, SNNs have the distinctive characteristics of binary output, high sparsity, and biological plausibil- ity. Therefore, compared with traditional ANN models, SNNs can further improve computational efficiency and reduce power consumption, which facilitates their remarkable superiority in the ap- plication of neuromorphic chips (Merolla al., 2014; Davies al., 2018: DeBole et al., 2019). Considering that an effective learning algorithm has not yet been found for SNNs, ANN-SNN con- version and backpropagation through time (BPTT) are still the two most commonly applied training methods. Compared with BPTT, ANN-SNN conversion provides a way around the nondifferentiable problem in the direct training procedure for SNNs and thus reduces the overall training complexity.
The aim in ANN-SNN conversion is to establish the mapping relationship between the activation output and the average firing rate. Traditional conversion methods exploit larger time-steps to over- come conversion errors and thus achieve high performance (Diehl et al., 2015). Many of the follow- ing works have attempted to optimize the performance from multiple perspectives, including using the soft-reset mechanism (Han et al., 2020), proposing more adaptive activation functions (Ho & Chang, 2021; Bu et al., 2022b), adopting a trainable threshold (Sengupta et al., 2019; Ding et al., 2021; Bu et al., 2022a), etc. However, these strategies cannot effectively eliminate the errors caused
"Corresponding author; yuzfl luzf12@pher 2@pku.ddu.c

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Published as a conference paper at ICLR 2023
by the deviation between the actual and desired firing rates, especially when the number of time- steps is small. Some recent works explore compensating for the errors by introducing burst spikes (Li & Zeng, 2022) and signed spiking neurons (Li et al., 2022). Unlike these works, our paper attempts to eliminate the errors with vanilla spiking neurons and answer the question of how to improve the performance of a converted SNN and possibly approach the upper bound performance.
In this paper, we observe and identify the source of conversion errors and propose an iterative opti- mization method based on shifting the initial membrane potential, which can fulfil accurate mapping between ANNs and SNNs under ideal situations. Our main contributions are summarized as follows:
We introduce the concept of offset spike to infer the deviation between the actual SNN firing rate and the desired SNN firing rate. We note that cases of firing one additional (or one less) spike are the main reason cause of conversion errors.
2 We propose a method to judge offset spike based on the residual membrane potential and an optimization method to eliminate conversion errors by shifting the initial membrane potential up or down. We derive the optimal shifting distance and prove that one spike can be increased or decreased under this condition.
We evaluate our methods on CIFAR- 10/100 and ImageNet datasets. The proposed method outperforms the existing oath--fftheeart ANN-SNN conversion methods using fewer time- steps. For example, we achieve 67.12% 12% top-1 accuracy on ImageNet with only 6 time- steps (4 time-steps for calibration and 2 time-steps for inference). Moreover, it is worth noting that we have reached the same level of performance as BPTT under the condition of significantly reduced memory and computing resources requirements.
4 We discover that our proposed method has an iterative property. Under ideal cireumstances, the deviation within the range of l spikes will be eliminated entirely after adopting our approach times. After iterations, the mean--quare error between the actual and desired firing rates of the output layer can reach 0.001 for the VGG-16 model on CIFAR-100.
2 RELATED WORKS
The principle of ANN-SNN conversion is map the parameters from pretrained ANN models to SNNs, which avoids training SNNs directly and reduces energy consumption significantly. The pri- mary goal match the ANN activation value and the average SNN firing rate. Cao et al. (2015) were the research pioneers in this field, and they replaced ReLU activation layers in ANNs with spiking neurons to fulfil the conversion procedure. Ho & Chang (2021); Bu et al. (2022b) pro- posed new activation functions, which better fit the finiteness and discreteness of the spike firing rate. Rueckauer al. (2017); Han et al. (2020) adopted "resetbb-subbracction" mechanism, which alleviated the problem of information loss and effectively improved the precision of the conversion process. For the setting of firing threshold, various strategies have been proposed, including Ro- bustNorm (Rueckauer al., 2017), SpikeNorm (Sengupta et al., 2019), and adjustable threshold (Han et al., 2020; Ding et al., 2021; Ho & Chang, 2021; Bu et al., 2022a;b), etc. Recently, spiking neural networks with high accuracy and low latency have become the focus and target of academic research. To reduce the time latency of the network, one must carefully address the exact spiking time of neurons. Deng & Gu (2021); Li al. (2021) fine-tuned the bias in each layer under the uniform current assumption. Nevertheless, the actual current would never be distributed uniformly. In terms of expectation, Bu et al. (2022b) proved that one-half of the threshold is the optimal value for the initial membrane potential, and that charging at this value can prompt neurons to spike more uniformly. However, as the authors pointed out: there is still mismatch between ANN and SNN due to the so called "unevenness error". In addition, other methods like burst spikes (Li & Zeng, 2022) and signed spiking neurons (Wang et al., 2022a; Li et al., 2022), have also been introduced to further improve performance. These efforts have aimed to alleviate the conversion loss. However, they undermined the biological plausibility and binary property of spiking neurons.
In addition to ANN-SNN conversion, backpropagation with exact spike time is another common way to train SNNs. The surrogate gradient (O' Connor et al., 2018; Zenke & Ganguli, 2018; Bellec et al., 2018; Wu et al., 2018; 2019; Kim & Panda, 2020; Zenke & Vogels, 2021) has been widely used to tackle the nondifferentiable problem in the training process, which substitutes the Heavi- side function with a derivable function. With the help of the surrogate gradient, hackpropagation
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through time enables the network to adjust weights by focusing every exact time-step. On this basis, Rathi & Roy (2021); Guo et al. (2022) further attempted the optimization of hyper-parameters and gradients. Bohte et al. (2002); Kheradpisheh & Masquelier (2020); Zhang & Li (2020) proposed timing-hased learning method, which viewed the specific spike firing time as significant temporal information to transmit between layers. Nevertheless, this type of method only applies to shallow networks at present. In addition, hybrid training methods have recently attracted extensive attention. Wang et al. (2022b); Rathi & Roy (2021) combined ANN-SNN conversion with BPTT to obtain higher performance under low latency. Kim et al. (2020) adopted rate-coding and time-coding si- multaneously to train SNNs with fewer spikes. Mostafa (2017); Zhou al. (2021); Zhang & Li (2020) established a linear transformation about the spike firing time from adjacent layers, which enabled the use of SNNs under the training mode of ANNs. In addition, BPTT can enable calibra- tion of the spike time in the training phase (Rathi et al., 2020). These works alter weights when training and stress the importance of spike timing, which is usually ignored in conversion methods. Inspired by these approaches, we incorporate the concept of calibration spike timing by manipulat- ing membrane potentials into the conversion pipeline to bridge the gap between ANNs and SNNs.
PRELIMINARIES
3.1
NEURON MODELS
For ANNs, the input a'-1 to layer is mapped to the output a' by a linear transformation matrix W and nonlinear activation function f(),, that = 1, L):
a' (
(1)
where f()) is often set as the ReLU activation function.
For SNNs, we adopt Integrate-and-Firg (IF) Neuron model (Gerstner & Kistler, 2002), which is sim- ilar to the approach reported in previous works (Cao et al., 2015; Diehl et al., 2015). To minimize in- formation loss during inference, neurons perform Freset-yy-subtraction 2020), which means that the firing threshold 6' is subtracted from the membrane potential after firing. The overall kinetic equations of IF Neuron can be expressed as follows:
mechanism (Han al.,
v' v'(t) v'( (t--tt()))))))) 1) r'(() s' (1)00,
('(t) = w w's'-1(1)0'--
(3)
Here v1( and (t) denote the membrane potential and input current of layer at the (-th time-step, respectively. W is the synaptic weight between layer 1-1 and layer and 0f is the spike firing threshold in the I-th layer. (() represents whether the spike fires at time-step l. For the i-th neuron, the current potential exceeds the firing threshold 6', the neuron will emit a spike. This firing rule can be described by the equation below.
4001 I, o/( (1t-1)+4()) 0' o, (()=))+(()<0
(4)
If not otherwise specified, the subscript 2; denotes the i-th element of æ.
3.2 ANN-SNN CONVERSION
The main principle of ANN-SNN conversion is to map the firing rates (or postsynaptic potential) of spiking neurons to the ReLU activation output of artificial neurons. Specifically, by summing equation from t to = T. and then substituting variable r'(t) with (t) using equation 3, and finally dividing T' on both sides, we obtain the following equation:
Σr. (1))0' W. ET sl (1)0011 (- ( (To 00 o0))
(5)
T
T
where T' denotes the total simulation cycle. For simplicity, we use the average postsynaptic potential 2 /T in equation 5, then we obtain
o' (T) as a substitute for the term Σ s(()))))
('(T)= w (' + (-r(())
x' (T) - -0)
(6)
T
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Layen!!
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(a)
(b)
(c)
(d)
Figure 1: The distribution of offset spike in each layer. (a) and (c): VGG-16 on CIFAR-10, (b) and (d): VGG-16 CIFAR-100, w/ constraint denotes the constraint of 2b' through the rectification of the spikes.
Equation can be approximated by a linear transformation between ø' (T) and 6' (T) as tends to infinity, which is exactly the same the forward propagation (equation 1) in ANNs due to ('(T) > This result implies that we can achieve lossless ANN-SNN conversion when tends infinity. However, the performance of converted SNNs degrades seriously under the condition of short time-steps 7 (Rueckauer et al., 2017; Han et al., 2020). To achieve high-performance SNNs under low latency, Bu al. (2022b) proposed replacing the commonly used ReLU activation func- tion of source ANNs with the quantization clip-floor-shift (QCFS) function:
X1
[W'a'- X
+ +1 2 ..L)
a' = =((a-))) clip
where L denotes the ANN quantization step and X is the trainable threshold of the outputs in ANN layer 1, which is mapped to the threshold 0' in SNN layer L. This paper follows the conversion
framework of Bu et al. (2022b) with QCFS function.
4 METHODS
In this section, first compare the outputs of ANNs and converted SNNs in each layer. We intro- duce offset spike to measure the degree of deviation between the actual firing rate and the desired firing rate in SNNs. Then, demonstrate that the offset spike of being one accounts for the main part in each layer and is the main reason of conversion error. Based on this, we propose suffi- cient conditions we present spike calibration strategy to eliminate conversion errors through shifting the initial membrane potential.
determine if offset spike exists and what the sign of offset spike value is, and
4.1 OFFSET SPIKE AND ITS DISTRIBUTION
ANN-SNN conversion errors can be divided into clipping error, quantization error (flooring error), and unevenness error (deviation error) (Bu et al., 2022b). In previous works (Han et al., 2020; Li al., 2021; Meng et al., 2022b), those errors are eliminated (or reduced) separately, and thus far no method to climinate the unevenness error (deviation error) has been identified. Since find
that the essential cause of most conversion errors comes from the remaining term r'(T)-n'(0) in equation 5, we consider reducing conversion errors directly based the prior knowledge of the remaining term. To measure the degree deviation between the actual firing rate and the desired firing rate, first introduce the definition of offset spike.
Definition 1. We define OFFSET SPIKE rb of layer as the difference between the desired total spike count C Cdssigned and the actual spike count C Caccual during the interval [0, T). that is
a'T 0r
/b Cdesigned
Cdl. actuual
5')).
(8)
where we set the maximum value X' of output a' in ANNs equal to the threshold 0l in SNNs, that a' denotes the normalized output in ANNs, which is mapped the firing rates of is, A' = 0¹. Thus,
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Ediry (eTi
2/ 0
3
5
3
Time step
Time step
Time-top
Timestop
(a)
(b)
(c)
Figure 2: Shifting up (down) the initial membrane potential can increase (decrease) one output spike.
SNNS, and C designed oi the gap between the actual and desired firing rate of the i-th neuron in layer of the SNN is & spikes. We further investigate the detailed ANN and SNN outputs in each layer. We train the source ANN with the QCFS activation function (equation 7) and then convert it to an SNN (more details are in the Appendix). Fig. (b) illustrates the distribution of offset spike for the converted SNNs with VGG-16 structure on CIFAR-10 10 and CIFAR- 100, respectively. We have the following observation.
a'r denotes the desired total spike count. Note that of = t indicates that
Observation 1. r/r 1 accounts for the main part in each layer and ep² = =3 rarely occurs.
Considering the cumulative effect of conversion errors the deep layer, the offset spike vp in layer I can be considered the joint effects of the offset spike rb' in layer - 1 and conversion erross in layer 1, and tends to increase with the increase in the number of layers.
For deeper analysis of the offset spike in each layer, we rectify ANN output in layer - make a = o' (T)) and rb'-1 = 0 (Sec. A.I I for more details of the constraint), and then compute the offset spike uf² in layer After the rectification for each layer, the distribution of offset spike for the converted SNNs with VGG-16 structure on CIFAR- 10 and CIFAR-100 are shown in Fig. I(c)-1(d), respectively. We have the following observation.
Observation 2. With constraint, 26 = +1 accounts for the main part in each layer and | > rarely occurs.
Observations 1-2 show that the firing of one additional (or one less) spike is the main cause of conversion errors, which implies that eliminate errors after adjusting Σ s' (t) with +1.
4.2 JUDGE CONVERSION ERRORS THROUGH RESIDUAL MEMBRANE POTENTIAL
Before we propose the optimal strategy for adjusting the output spikes to eliminate the offset spike w¹, we need to determine the offset spike exists and what the sign the offset spike value is. the sign of the offset spike is positive, which corresponds to the situation in which the ANN output larger than the SNN output, the spiking neurons should fire more spikes to eliminate the offset spike, otherwise, they should fire fewer spikes.
In the practical application of SNNs, we cannot directly obtain the specific value of the offset spike Tb.. Fortunately, we find that can determine the sign of dp² according the value of the residual membrane potential v'(T), We have the following theorem:
Theorem 1. Suppose that an ANN with QCFS activation function (equation 7) is converted to an SNN with T. = T, X = 0¹ x² (0) 02,2, and the imputs to the 1-th layer of the ANN and the SNN are the same, that is, al-1
(T)) Then for any i-th element of the 1-th layer, we can draw the
following conclusions:
(i) If of (T") > 0 and oi (T) < O, we will have (.(T) > d; and v < 0. (ii) If o (T) < 6¹ and of (T) > 6¹, we will have (T) < af and v > 0.
The proof is provided in Appendix. (i) implies that if the postsynaptic potential is larger than 0 and the residual membrane potential smaller than O, we can conclude that the neuron fires more spikes than expected and the sign of the offset spike value is negative. (ii) implies that if the postsynaptic potential smaller than and the residual potential is larger than 6, we can conclude that the spiking neuron fires fewer spikes than expected and the sign of the offset spike value is positive.
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4.3 ELIMINATE CONVERSION ERROR THROUGH SHIFTING INITIAL MEMBRANE POTENTIAL
Since the firing of one additional (or one less) spike (ep" - 1) is the main cause of conversion errors, we propose an optimization strategy to rectify the output spike Σ s' (t) by adding or subtracting one spike, thereby eliminating errors. Specifically, we consider adjusting the value of Σi=1 st( by shifting the corresponding initial membrane potential v (0) up or down. One intuitive explanation is that higher initial membrane potential will make the spiking neurons fire earlier and will increase the firing rates during the period [0, 7), while a lower initial membrane potential will make the spiking neurons fire later and will decrease the firing rates. The following theorem gives the optimal shifting distance when we attempt to move Σ s' (f) (and w by a distance of =1.
Theorem 2. If we use (t) and (t) denote the binary spike of the i-th neuron in layer at time- step before and after optimization, (0) and (0) to represent the initial membrane potential before and after optimization, then Ve E (0, 0'), will have the following conclusions:
(i) we set (0) (0) max(0', min(f =) +3++20 e), then Σ 31(0 Σ
81(1)= -1
[=1
T
(ii) we set (0) = (0) - max(f' 0 + -
mmax()(())ss(( + = 0})) then Σ (t) = Σ +1(t)+1.
The proof is provided in Appendix. Note that the variable e illustrates that as long as the initial membrane potential is within a certain range, the number of output spikes can be guaranteed to increase or decrease by
Example 1. Fig. 2 shows four different scenarios before and after shifting o (0) that verify the ef- fectiveness of our theorem. Specifically, Fig. ((b)-2(b)/Fig. 2(c)-2(d) indicates two cases of shifting down/up, which correspondes to (i)((i) in Theorem 2.
By combining Theorems and 2, we propose the complete spike calibration algorithm. Our method can be divided into stages. First, for 1-th layer, we spend p time-steps to determine the specific spike firing situation. According to Theorem 1, if of ( < 0 (or (p) > 0¹), by combining (p),, we can infer that of (p) is actually larger (or smaller) than the expected average postsynaptic potential a!, and vif < (or v > 0). In addition, we will preserve the membrane potential after each time-step, which will be used to calculate the subsequent optimal shifting distance.
In the second stage, based on Theorem 2, we will calculate the optimal shifting distance of the initial membrane potential for specific neurons with conversion errors. Generally speaking, if v/ < 0, we will shift its initial membrane potential down by calculating (i) from Theorem 2, if w/ >> 0, we will shift its initial membrane potential up by adopting (ii) from Theorem 2. After optimizing the initial membrane potential, we will spend T time-steps implementing the test on corresponding datasets and deliver the output to the I + 1-th layer.
4.4 ITERATIVE PROPERTY OF OUR OPTIMIZATION METHOD
In the previous section, we show that shifting the initial membrane potential up (or down) can change a case of s² = 1 to xb² = 0. In fact, our method also converts the case of ub² = + to eb' = + 1). As long the offset spike qb' is not zero, the performance of converted SNNs will degrade. One important problem to address is whether we can further eliminate the offset spike in situations where x/ > 2.
Fortunately, we find that our optimization method has an iterative property. One can reuse Theorem 2 to increase (or decrease) one output spike each time. Of course, it comes at a significant compu- tational cost. In the Experiments Section, we will show that the performance of the converted SNN increases with the iteration. Typically, we can achieve reg-pperforance and low-latency SNNs with only one iteration.
5 EXPERIMENTS
In this section, we choose image classification datasets to validate the effectiveness and performance of our proposed methods, including CIFAR-10 (LeCun et al., 1998), CIFAR-100 (Krizhevsky al.,

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Figure The distribution of offset spike before and after optimization
Table I: Comparison with existing state-of-the are ANN-SNN conversion methods
Method
ANN
Architecture
T=1
T=2
T=4
T=8
T=16
T=32
CIFAR-100 Dataset
SNM 74.13%
71.80%
SNNC-AP
77.89%
73.53%
OPI
76.31%
VGG-16
60.49%
70.72%
74.82%
QCFS
76.28%
63.79%
69.62%
73.96%
76.24%
77.01%
Ours
76.28%
74.24%
76.03%
76.26%
76.52%
76.77%
76.96%
RMP
68.72%
27.64%
70.43%
23.09%
$2.34%
67.18%
ResNet-20
QCES
69.94%
19.96%
34.14%
55.37%
67.33%
69.82%
69.97%
59.22%
64.21%
68.18%
67.17%
69.445
70.29%
ImageNet Dataset
SNNC-AP
75.36%
63.644 63.64%%
SNM
73.18%
64.78%
OPI
74.85%
VGG-16
6,23%
36.02%
64.70%
QCFS
74.29%
19.12%
50.97%
68.47%
Ours
74.19%
63.84%
70.59%
72.94%
73.82%
74.09%
74.33%
SNNC-AP
75.66%
64.54%
QCES 74.32
ResNet-34
35.06%
59.35%
69.37%
Ours
74.22%
69.11%
72.66%
73.81%
74.17%
74.14%
73.93%
2009) and ImageNet (Deng et al., 2009) datasets. The network architectures selected for evaluation include VGG-16 (Simonyan & Zisserman, 2014), ResNet-18, 18, ResNet-20 and ResNet-34 (He et al., 2016). For the setting of the hyperparameter p. we set p 4 for CIFAR-10/100 and p for ImageNet if there are no special instructions. More details of the experimental settings are provided the Appendix.
5.1 EFFECTIVENESS OF THE PROPOSED METHOD
To illustrate the effectiveness of our proposed initial membrane potential shifting operations, compare bar charts offset spike e each layer of SNNs before and after the shift. Fig. illustrated the results of VGG-16 networks the CIFAR-10 10 and CIFAR-100 datasets. It can be observed that the shifting operations significantly reduce offset spike, that is, the deviation between ('(T) and at, for each layer. For vanilla settings without the shift operation (denoted "w/o shift" in Fig. 3), one can discover a magnification effect of spike count error from the 1st to the 11th layer. In contrast, the apparent magnification is alleviated with the proposed methods. From Fig. 3(b), we notice that the #2 and =3 offset spike in the VGG-16 16 model for CIFAR- 100 have increased compared to those for CIFAR-10 (Fig. 3(a).. Using our method can clearly decrease these deviations and achieve a comparable error-free conversion.
5.2 COMPARISON WITH STATE-OF-TIIB-ARI METHODS
We compare our methods with previous state-of-the the-rtt art ANN-SNN conversion works, including RMP (Han et al., 2020), SNM (Wang et al., 2022a). SNNC-AP (Li et al., 2021), OPI (Bu et al., 2022a), QCFS (Bu et al., 2022b), on CIFAR-10, CIFAR- and ImageNet datasets. Since we spend time-steps in the first stage acquire relevant temporal information about membrane potential,

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Table Comparison with other types of SNN training methods
Dataset
Method
Type
Architecture
Accuracy
Time-step
Dual-Phase
Hybrid Training
VGG-16
70.08%
Diet-SNN
Hybrid Training
VGG-16
69.67%
CIFAR-100
RccD
BPTT
VGG-16
69.88%
5
Oursip 4)
ANN SNN converston
VGG-16
74.24%
HC-STDB
Hybrid Training
ResNet-34
61.48%
250
DSR
Supervised learning
PreAct-ResNet-18
67.74%
50
STBP-tdB
BPTT
ResNet-34
63.72%
ImageNet
PLIF
BPTT
ResNett-3
67.04%
TET
BPTT
ResNet-34
64.79%
6
Ours 4)
ANN-SNI conversion
ResNet-34
67.12%
Table 3: The ratio and MSE after multiple iterations
Baseline
Ours
Ours
Ours x
Dataset
Architecture
Ratio
MSE
Ratio
MSE
Ratio
MSE
Ratio
MSE
VGG-16
88.33%
0.120
97.65%
0.024
99 83%
0.002
99.84 19.84%
0.002
CIFAR-IA 10
ResNet-28 20
62.38%
0.512
82.56%
0.179
99.73%
0.003
99.764%
0.002
V6G-16
82,90%
0.192
98.42%
0.016
99 86%
0.001
99.87%
0.001
CIFAR-100
ResNet-20
41.59%
1.641
67.08%
0.453
86.03%
0.165
91.29%
0.101
will compare the performance of other works at time-step T + p with our performance at time-step to ensure the fairness of comparison.
Tab. reports the results on the CIFAR-100 dataset. For VGG-16, our method at time-step 1 (p 4) outperforms SNM and SNNC-AP at time-step step 32. Moreover, we achieve 76.26% top-1 accuracy with 4 time-steps (p 4), which is 2.30% higher than QCFS (73.96%, T=8) and 15.77% higher than OPI (60.49%, T=8). For ResNet-20, the performance of our method at time-step 1 (p = 4) surpasses the performance of RMP time-step 32 (59.22%
27.64%). The accuracy of our method is 18. at time-step (p = 4), whereas accuracies of QCFS and OPI are 55.37% and 23.09% at time-step 8, respectively. More results CIFAR-10 are listed in the Appendix.
We further test the generalization of our method on the ImageNet (Tab. 1). For 16, we achieve 73.82% top-1 accuracy at time-step 8 (p 8), which outperforms QCFS (50.97%, T=16) by 22.85% and OPI (36.02%, T=16) by 37.80%. For ResNet-34, our method at time-step (p = 8) outperforms SNM and SNNC-AP time-step 32. Moreover, we achieve 74.17% with time-steps (p 8), which is 14.82% higher than QCFS (59.35%, T=16). These results show that our method can achieve better classification accuracy with fewer time-steps.
0.7
Accunccy
0.75
0.70
Accuney
0.6
Accumcy
Acoumey
0.6
0.7
0.65
0.68
0.60
0.5
0.64
0.5
8
16
2
16
2
16
Time-step
Time-stpp
Time-step
Time-stop
(a)
(b)
(c)
(d)
Figure 4: Influence of different (a) VGG-16 on CIFAR-100, (b) ResNet-20 on CIFAR-100, (c) VGG-16 on ImageNet, (d) ResNet-34 on ImageNet.
In addition, we compare our method with other types of SNN training methods (Hybrid Training & BPTT), including Dual-Phass (Wang et al., 2022b). Diet-SNN (Rathi & Roy, 2021), RecDis-SNN (Guo et al., 2022), HC-STDB (Rathi et al., 2020), STBP-udNN (Zheng et al., 2021), PLIF (Fang et al., 2021), TET (Deng et al., 2022) and DSR (Meng et al., 2022a). Here we set p 4 for the CIFAR- 100 and ImageNet datasets. As reported in Tab. 2, our method achieves better accuracy on the CIFAR-100 100 dataset and comparable accuracy the ImageNet dataset with the same quantity of time-steps. Note that compared to ANN-SNN conversion, the bacck-rroagation approaches need to propagate the gradient through both spatial and temporal domains during the training process,
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Re10.
0.6
0.8
1.5
menn [|gh|]]]]
0.4
mei ||
0.6
mocol [
1.0
10:0m
0.1
0.2
0.4
0.2
0.5
0.0
0.0
0.0
0.0
Layen!
Layeni
Layeni
Layer
(a)
(b)
(c)
(d)
Figure 5: The MSE of conversion error after using iterative optimization. (a): VGG- 16 on CIFAR-
10, (b): ResNet-20 on CIFAR-10, 10, (c): VGG-16 CIFAR- 100, (d): ResNet-20 CIFAR-100.
0.4
Percentage
Percentage
3.0445
0.3
0.2
0.1
0.15
12
15
18
12
15
188
Layeri
Layerét
(a) ResNet-20 on CIFAR-10
(b) ResNet-20 on CIFAR-100
Figure 6: The distribution of offset spike after using iterative optimization
which consumes large amounts of memory and computing resources. All these results demonstrate the superiority of our method.
5.3 EFFECT OF THE INFERENCE TIME-STEP P
We further explore the influence of the hyperparameter p in the first stage of our method. Fig. shows the accuracy of the network with different values of For Fig. ((a,4(d)),(d), the value of quan- tization level QCFS function set to 4, 8, 16 and 8. We find that the SNN accuracy tends to converge P gradually approaches L. This phenomenon can be understood as follows. We use the QCFS activation function in the source ANN and we have a' E (ko' /Ljk = 0, 0,.,... L} and o' (T)) E {ko' /plk 0, 1,..., p}. Thus, the mapping relationship between a' and (T(T) will be- come more accurate when p approaches L, which makes the temporal information obtained from the first stage more precise to improve the performance of the network.
5.4 EFFECT OF THE ITERATIVE OPTIMIZATION
In section 4.4, explain that method has iterative property that can reduce the offset spike through multiple iterations. To demonstrate this, we define the ratio as the percentage a = o (T) in the output layer. Despite this, we also consider the indicators of equan- error (MSE), which is defined as || ||3- Tab. reports the ratio and MSE of the output layer, in which the baseline iterations. Besides, we set L T. From top to bottom in Tab. 3, the values of L are set to 4, 4, 4 and 8. From Tab. and Fig. 5, we can conclude that, generally. the ratio and MSE each layer continue to decrease as the number of iterations increases, which is consistent with the results shown in Fig. 6.
denotes the performance without using our methods and x2 represents
6 CONCLUSIONS
In this paper, we first define offset spike to measure the degree of deviation between the actual and desired SNN firing rates. Then we analyse the distribution of offset spike and demonstrate that we can infer the specific value of the deviation according to the corresponding residual membrane potential. Furthermore, we propose an optimization method to eliminate offset spike by shifting the initial membrane potential up and down. Finally, demonstrate the superiority of our method on CIFAR- 10/100 and ImageNet datasets. Our results will further facilitate the relevant research and application of SNNs neuromorphic chips.
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China under Grant No. 62176003 and No. 62088102.
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A APPENDIX
A.1 THE NETWORK CONFIGURATION IN THE TRAINING PROCEDURE
We choose Stochastic Gradient Descent optimizer (Bottou, 2012) and Cosine Annealing scheduler (Loshchilov & Hutter, 2017) to train ANN models for 300 epochs. For CIFAR-10/100, the value of weight decay is set to 5 10 10-4, and the initial learning rates are 0.1 and 0.02, respectively. For 0.1 and weight decay
ImageNet, we set the initial learning adopt data-augmentation techniques (DeVries & Taylor, 2017; Cubuk et al., 2019; Li et al., 2021) to further improve the performance of the models.
10-4, In addition,
In Fig. 1, we train the source ANN with the QCFS activation function (equation 7) and then convert 4. For Figure I(c)-I(d), add the constraint that the output a'-1
an SNN. We T in layer I - 1 of ANNs is the same as the output Ø² (T) of SNNs, that is, a a-1 = Φ (T). and compute the offset spike with equation 8 and a' f(W' Φ 1-1 (T)).
A.2 PROOF OF THEOREM
Theorem Supposing that an ANN with QCFS activation function (equation 7) is converted to
an SNN with L = T.X = 0¹, v0) = 0¹ /2, and the inputs to the 1-th layer of ANN and SNN are
the same. that is, a'-1 60 (T). Then for any i-th element of the 1-th layer, we will have the
following conclusions:
(i) If th (T) > O and vi (T) < O, we will have (. > df and v < 0.
(ii) If (T) < 0f and (T) > 0', we will have (T) < at and v 0.
Proof. According to the preconditions and equation 5, we have:
Σ I(()
(T) 0'/2
d(T) = T
(SI)
T
T
Σ T1(1) El 61-00/2,0'' + 0 /2), based on the preconditions and equation 7, we get:
Σ
(t)
0
1
a.
+
(S2)
T
8
2
T
When Σ (tt e [k0t 82/2, k0 + 0 02/2), = 0, 0.1,..., T. from equation S2 we will have af = 1=1
k0' /T. For (i), by combining (T) < and equation S1 we will have:
Σ (t)
- 0
((T)
T
T
T Σ
(t 0 //TTT
/T d
(S3)
When Σ T/(() -0'/2, 0, according the precondition (T O, have ({ of- T
In addition, if Σ T((1) 0T + e 01/2, according to v/ (7) < 0 and equation $1, ((( > 0*, which
1=1
impossible. Therefore, we can derive that (T) > a of = (a) - ((())T// < and we have
already proved (i).
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For (ii), we also first consider Σ (
-1 When Σ 1(0)) E 1000-0'12 1/22kk k0' +
0 /2 k = 0,1, ...., T, from equation S2 we will have of kf /T, by combining uT(T) > 0 and
equation SI, we will have:
T [()
Σ
12
(T)
=1
T
T
[(t/TT - 0 /2T.
k0 a
(S4)
When (()) > 0 = 0', according to the precondition of(T) < 0', we have (T
<
T Σ (t) -8 /2. according (T) > o and equation S1, (T) < 0, which
a. addition, if impossible. Therefore, we can derive that (T) < af, of = ((T))T/0 > and we have proved (ii).
Theorem 2. If we use (L) and (1) to denote the i-th element in the binary oulput of the 1-th layer at time-step before and after optimization, (0) and (0) to represent the initial membrane potential before and after optimization, then Ve (0, 0¹), we will have the following conclusions:
(i) If we set (0) = (0) - max (0', min (o((0))s((t) = 13 + e). then Σ 3 = -
T
- $1(1) 1
(ii) we set (0) = (0) + max (0 0 + max (v((0)s((1) = 0}). then Σ( Σ s(1)+
We use (t). (t) to represent the accumulative potential at time-step t before and after using optimization. Before the proof of theorem, we firstly introduce Lemma 1.
Lemma 1. For situation (i) in Theorem 2, 3t [1, E ((K) E ({(k) + 1. For situation (ii)
in Theorem 2, 3t E [1,7], Σ s((R)) Σ (k) 1
Proof. For situation (i) in Theorem 2, we use to to denote the specific time when ef(t))
min (1))(() = 1} A ((()))) = 1. VI, as we optimize SNNs layer by layer, we have the fol-
lowing equation:
m!(() (0) Σ ((K))
(S5)
ml()) = 50 + Σ ((())
(S6)
As 20( > (0) and we use the same input Σ I((k) before and after optimization, when
Σ ((k))' (k))) we will have m((1) > m)()) and further derive st(() > ((),,, which
v Σ ((())0)
means that Vt, Σ (k)))) >
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Σ s((()) E (k) 1, then have already found qualified time t'. If
If 3t (1,
Σ (()) Σ (k)), we will have:
1=1
(to -m((to) = -0 27(0)
max (0', min (v((0)s:(() 1) +c)
max (0', (((((())+
(S7)
As (ttt)) = et((t) + 0¹, we will further have:
ml(lo) = mm((l) max (0', (b(()))) +
< <mi(to) 11(10)) -
8
(S8)
From the above equation, we can derive m!(to) < 0 and st())) (t,)) then we will have Σ ((()) = Σ (1)) + 1, which means that to is qualified time.
For situation (ii) in Theorem 2, we use to to denote the specific time when (1()))
max (t) 0} A ((())) o. Similarly, we will derive Vt, Σ (k)) Σ (k()) according equation S5-equation S6.
k=1
Σ (((k)) E (k)) 1, then we have already found qualified time t'.
If 3t [1,to);
k=1
Σ (k)) = Σ (k)), we will have:
ml((t) ml((t)) = (0) - e0)
max (0 0¹ max (o((0)(((()) = 0))
- v/((o)). max (0'00' -
(S9)
As m((()))) = v((t)), we will further have:
ml(l.)) m((( + max (0', 0 + - v((()))
> ml(1o) + 0f + -
> 0..
(S10)
From the above equation, we can derive m.(t.) > 0¹ and st(.t) = 0, (to)) = then we will have (((k)) - Σ (b) 1, which means that to is qualified time.
Now we will further prove Theorem 2.
Proof. Proof of (i). 1f min 1)
d((0) 0¹. According Lemma 1, +1) then we will have m/(ts +1) = m (t. ft+ +1) by combining equation $5
St. Σ (k) E k=1
T
T
and equation S6, which means that
Σ
s((k) = Σ
(k(k)) in the remaining time cycle. As
k=t+t1
k=t-1
T result, we can have Σ
s((()) = Σ (1)) +1.
k=1
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If < min (t((t)ss = 1) + (0 = 2/(0)- min
min (o(())s((() 1 - According to Lemma 1, 2. Σ s(()) Σ (k) 1, then we will have m/ +1 = (t + 1) + min (o((t))s((t)
to
k=1
- 0¹, which means that (t. + +) > mt((s + 1).
For m if we set t' as the first spike firing time after t, which means that m)(t) o/(() 0
and Σ (k) = Σ ((()) + 1, then we will have (t) = m(((') - min (o((0)oo((1) = 1]
r'11
e c+ 00 = (t((t) min (t)(t)ss((t) = 1} - - 20' > 0¹. which means that si(t) = 3{((') =
1, E. (k)) Σ (1(+ 1. If we continue to use the above derivation process, we can finally have
k=1
T
T
Σ ((k)== = E 31() +1.
k=1
k=1
Proof of (ii). lf > l + - max (o((t))s((t) = 03, 2( = (0( + 6'. According to Lemma 1,
3t, Σ (k) Σ (k) 1, then we will have m (t +1) = mm (t, +1) by combining equation S5
k=1
T
T
and equation S6, which means that
Σ
(k))
Σ
(1())) in the remaining time cycle. As a
k=
1=t-11
Σ (kk) = T Σ (k) 1.
result, we can have
k=1
If el 0 + max (t) (t) = =0. (0) (0) 0 + max (u/(t)(s((t) = 0}. According
to Lemma 1, 3t., Σ (k( - Σ (k) then we will have m!(t. + 1) m/ 1) +
max {c((t)[s(((t) = 0}. which means that (t. + 1) < (t + 1).
For m/. if we set t' as the first spike firing time after t,, which means that m!(t) (/((²) + A
and 1'1 8' Σ (k) - 1. Similar to situation (i), we need to prove that s/((') = (t) = 1.
(() =
However, it is not easy to make a direct proof. Therefore, we attempt to prove its inverse and negative
thesis: : when t' > t A Σ ( = Σ (1))) - 1, if s!() = 0, then we can have (((()) = 0.
Under this condition, we can derive m!((') = u/ (t')
m((() - max (c(())ss(() 0}.
then we will have (t') = (((()) + max {u) (t) = 0}. As max (=((t))(((t)= = 0) > (tt)) m(((t) & E < 0', which means that (t') = 0.
Therefore, if we set t' as the first spike firing time for m. after t., we can prove that st(t') = -t)
1, which means that
(k) Σ (A)) 1. If we continue to use the above derivation process,
T
we can finally have Σ (((k)) = Σ ({(k) - 1.
A.3 EXPERIMENTAL RESULTS ON CIFAR- DATASET
Tab. SI reports the results on CIFAR-IO dataset. For VGG- the accuracy of our proposed method is 95.46% with 4 time-step (p = 4), whereas the accuracies of OPI and QCFS are 90.96% and 94.95% with 8 time-step. respectively. For ResNet-18 18, we achieve 95. 464 with 4 time-steps (p- = 4), whereas the corresponding performance of OPI and QCFS are 75.44% and 95.04%. For ResNet-20, our method reaches 91 .88% with time-steps (p = 4), which is 2. 13% higher than QCFS (89.55%, T=8) and 25.44% higher than OPI (66. (66.24%, T=8).
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Table S1: Comparison with other ANN-SNN conversion methods on CIFAR-I0 dataset
Method
ANN
Architecture
T=1
T=2
T=4
T=8
T=16
T=32
SNM
94.09%
93.43%
SNNC-AP
95.72%
93.71%
OPI
94.57%
VGG-16
90.96%
93.38%
94.20%
QCFS
95.52%
91.18%
93.96
94.95%
95.40%
95.54%
Ours
95.51%
94.90%
95.36%
95.46%
95.51% %
95.57%
95.61%
SNM
95.39%
94.03%
SNNC-AP
95.46%
94.78%
OPI
96.04%
ResNet-18 18
75.44%
90.43%
94.82%
QCFS
95.64%
91.75%
93.83%
95.04%
95.56%
95.67%
Ours
95.64%
95.25%
95.45%
95.46%
95.66%
95.68%
95.68%
OPI
92.74%
66.24%
87.22%
91.88%
QCFS
91.77%
ResNet-20
73.20%
83.75%
89.55%
91.62%
92.24%
Ours 91.77%
89.88%
91.26%
91.68%
91.86%
92.20%
92.16%
A.4 ELIMINATING OFFSET SPIKE THROUGH ITER ATIVE OPTIMIZATION
In Sections 4.4 and 5.4, we have pointed out the iterative property of our proposed method. Here we will make a discussion in detail. Firstly, we can infer the specific value of et based on the residual membrane potential when the corresponding input current belongs to a specific interval, which is illustrated in the following theorem.
Theorem 3. Supposing that an ANN with QCFS activation function (equation is converted to an SNN with L = T. x 0% v0( = 0¹ /2, and the inputs to the 1-th layer of ANN and SNN are the same, that is, al-1 = -b- Then for any i-th element of the 1-th layer, we will have the following conclusions:
T
If Σ Tt( 1-0' + 12), when v/(T)/0 [k, + 1), we will have 1ft = aTT//'
(1) k where Z.
Proof. As the preconditions of Theorem same as the preconditions of equation SI and equa- tion S2, by combining equation S1 and equation S2, we will have:
(()) [=1
Σ
1(t)
T
((T)) 0¹/2
T/ Σ
1001
2
1(1)/0' + +1/2 - E140/00 + +1/2)
T
(/TT//O' +
Σ
(SII)
Σ 1(11/0' +1/2 (5 1(1/0' + 1/2
T
As -
< 0. when (T) E [k + 1), k
dT//0' Σ (t()) k ++1. Considering that /
Σ st()) Z. we have os dTT/0'
Σ ((1) = k.
In fact, even if the input current does not belong to the specific interval, from equation 7, we can T
T derive that when Σ r'()) < - 0¹ 02/0./202222 a' 0 and when Σ (())) > 0'T + 0² 0-/2, 22,a al 0², then we
17

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Published as a conference paper at ICLR 2023
Table S2: Input/Output Ratio for each layer of an SNN with VGG-16 on CIFAR- dataset
Condition
1.3
155
L10
1.12
L14
Input Rutio
99.999%
99.97%
99.90%
99.63%
99,2415
95.775
97.84%
97.43%
57.2315
97.311
97.528
97.119
97.58%
97.50%
Oripae Ratio
99 999
99 999
999
99 995
99.9996
909
99
99.95%
9496
99.
09.00%
90 6878
99.
99,896
99.776
Unipur Katio when
Inpur accurate
1006
100%
100%
100%
100%
100%
100%
100%
1005
100%
100%
100%
100%
100%
1008
Table S3: Comparison with different initialization strategies
Method
Dataset
Method
T=1
T=2
T=4
T=8
T=16
CIFAR-100
VGG-16
Random Intialization
68.03%
74.74%
CIFAR-100
VGG-16
(0) + /2/2
63.79%
69.62%
73.96%
76.24%
CIFAR-100
VGG-16
(0) + (p),p=
73.38%
74.53%
75.09%
76.27%
76.62%
CIFAR-100
VGG-16
Ours(p= = 74.24%
76.03%
76.26%
76.52%
76.77%
can also directly determine the vs' according to the value of o' (T). After we have already acquired the value of vb", we will adopt our optimization method for | times eliminate the offset spike on i-th element neuron of the 1-th layer.
In Tab. 3, the Ratio after multiple iterations does not achieve 100%. We find that the non-zero MSE and Ratio in Tab. 3 are caused by the rounding of the floating-point numbers. Specifically, we carefully checked the Ratio, defined as the percentage of SNN input (output) equals ANN input (output) in each layer, to prove this, and we list the results in Tab. S2.
We find that the Ratio of the output in layer I is 100%, but the Ratio of the input in layer 2 is close to 100%. Thus, the error must be caused by the floating point number precision problem in multiplica- tion and division operations involved in the forward propagation between layer I and layer Con-
sidering that SNNs will calculate Σ W W') but ANNs will calculate W ( (t)/T) as
the average input current for the 1-th layer, these two corresponding inputs not necessarily equal due to the rounding of the floating point number.
We then conduct another experiment to prove that conversion errors can be reduced zero if the rounding of the floating point number is eliminated. We force the input of spiking neurons to be the same as QCFS neurons in each layer and calculate the Ratio of the output. As shown in Tab. S2 (line 4), we find that the Ratio of the output in each SNN layer is 100%, which indicates that iterating the proposed method can finally reduce conversion error to zero.
A.5 COMPARISON WITH DIFFERENT INITIALIZATION STRATEGIES
We make a comparison among different initialization strategies on CIFAR-100 with VGG-16 struc- ture, including random initialization, setting e0()) = 0¹ /2 (Bu et al., 2022b), using the residual membrane potential o' (p) of the first stage as the initial membrane potential and our proposed method. As shown in Table S3, our proposed method outperforms other initialization strategies under low time-steps, which proves the superiority of our method.
From Tab. S3 (line 4), we notice that using the residual membrane potential (p) as the initial mem- brane potential also achieves considerable performance. Therefore, besides our proposed method, we can also provide a lightweight optimization scheme: for each layer, we can consider directly selecting the residual membrane potential n' (p) after o steps as the initial membrane potential for our second stage. The idea is to make v (TT) o'( in equation 5 approach O to eliminate con- version errors (offset spike). Tab. S4 reports further results on the ImageNet dataset. Although the performance of our lightweight optimization scheme is weaker than our best solution, it is still much better than the current SOTA methods. Under the condition of using the lightweight scheme, we can avoid the extra calculation of the optimal shifting distance. We compare the running time among QCFS, our lightweight scheme, and our shifting method on CIFAR- 100 with VGG- 16 structure and 16 time-steps. The corresponding running time is 101s, 101s, and 134s, respectively.
18

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Published as a conference paper at ICLR 2023
Table S4: Comparison with the state-of-the-art ANN-SNN conversion methods
Dataset
Architecture
Method
T=1
T=2
T=4
T=8
T=16
OPI
6.25%
36.02%
QCFS
19.12%
50.97%
VGG-16
lightweight(ours) 62.27%
69.69%
72.50%
73.39%
74.04%
ImageNet
shifting(ours) 63.84%
70.59%
72.94%
73.82%
74.09%
QCFS
35.06%
59.35%
ResNet-34
lightweight(ours)
69.04%
69.63%
69. 99.80%
69.77%
70.97%
shifting(ours)
69. %
72.66%
73.81% 1
74.17%
74. 14%
A.6 PSEUDO-CODE FOR OVERALL ALGORITIIM FLOW
Algorithm 1 Algorithm for ANN-SNN conversion.
Require:
The quantity of time-steps to calculate residual membrane potential The quantity of time-steps to test dataset T; The iteration number the optimization strategy ItNum; The corresponding input for SNN layer data'; The shifting variable mentioned Theorem e; Pretrained QCFS ANN model SAN A); Dataset D).
Ensure: SNN model SNNN W, 0, 10 8).
1: # Convert ANN to SNN
2: for 1o 1 to fANN.lay layers do
3:
fs fANN.X X
4: fsNN. (0) = y JSNN. 8
5: fsNN.W* w = fANN. W
6: end for
7: # Eliminate offset spike
8: for (Image, label) D do
9: for = 1 to T do
data (1) = Image
11: end for
12: for / = 1 to fsNv.layers do
for epoch 1 to ItNum do
14:
Acquire the residual membrane potential for 1 1 p do
16:
SiNN.s'((pponl - 1) p + SSNN (data end for
17:
18:
# Optimize the initial membrane potential with the optimal shifting distance
19:
Need to shift the initial membrane potential up according to Theorem 3 then
p) fsNN. ('epocch - 1) max fanN/ (f)] (t)) = 0, E [(epoch - 1) x + 1, epoch x p]}) end
fSNN. v' (epoch
p) max(0' O¹
22:
if Need to shift the initial membrane potential down according to Theorem 3 then
23:
[SNN- (epoch x p) /SNN. ((eooc - 1) x () max (0', ,e + min (/SNN. ([)),sNN
o (1) = 1, [epocc 1) p + 1, epoch p)))
24:
end end for
25:
26:
for t = to do
27:
fsNN-8' (ItNum p t) fiNN (dattd (t)) data" (t) fSNN. W (fSNN.s' (ItNum p t) 0')
29: end for
30: end for
31:
end for
32: return SSNN W, 0, u,
19


Reviews:
[{"review_id": "eyl1rgo0lP", "reviewer": "Reviewer_oG9M", "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": "", "summary_of_the_paper": "The authors address the issues underlying ANN to SNN conversion and identify the firing of one more or less spikes as the main reason of failing conversion. This can be exploited in an optimization procedure over existence of an offset spike. Evaluation is done on simulated firing rates of cifar-100.\n", "summary_of_the_review": "The paper has very good technical content and it would make an acceptable contribution if the authors would write the non-mathematical part from scratch.\nI apologize that I cannot give a longer review, but the paper is not comprehensible.\n\nAuthors have made a significant revision and their efforts are really appreciated. ", "strength_and_weaknesses": "+: The paper has a good introduction to the ANN-SNN conversion paradigm\n+: The authors make two good observations about the offset spike\n+: The authors state and show two theorems. While the math seems right, it is difficult to judge their relevance  \n\nUnfortunately, the paper is pretty much unreadable and I could only understand the technical part following the equations only but not the text.\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": "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": "Ibv6NdYf5dw", "reviewer": "Reviewer_EoJ9", "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": "", "summary_of_the_paper": "The paper proposes an efficient ANN-SNN conversion method based on Spike Calibration. The experiments show that the conversion using the proposed approach reaches competitive accuracy with few time-steps.", "summary_of_the_review": "Borderline paper where several concerns should be clarified.", "strength_and_weaknesses": "Strengths:\n1. The tackled problem is relevant to the scope of ICLR.\n2. The results look promising.\n\nWeaknesses:\n1. It is recommended to change the title to avoid confusion with the following paper:\nYang Li, Xiang He, Yiting Dong, Qingqun Kong, Yi Zeng. Spike Calibration: Fast and Accurate Conversion of Spiking Neural Network for Object Detection and Segmentation. CoRR abs/2207.02702 (2022).\n2. In Section 2: \"Inspired by these works, we incorporate the concept of calibration spike time into the conversion pipeline to bridge the gap between ANNs and SNNs.\" This sentence is too vague. The differences between the related works and the proposed work should be properly discussed in detail.\n3. All the figures are too small (in particular the legends). It is recommended to increase their size to make them readable.\n4. The description of the proposed method in Section 4 looks too vague and hard to follow. It is recommended to use algorithms, schemes, examples to ease the discussion. (To save space, the content of Section 3 can be moved to the appendix.)\n5. It would be useful to provide the source code for reviewers' inspection during the rebuttal.", "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": "IC_AWnwV3NP", "reviewer": "Reviewer_NhRe", "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": "", "summary_of_the_paper": "This work proposes a spike calibration algorithm for reducing the conversion errors in ANN-SNN conversion models. The authors provide a new measurement for the conversion error by the offset spike and develop an optimization strategy that can shift initial membrane potential to offset the conversion errors iteratively. They also analyse the optimal shifting strategy theoretically. Their results on CIFAR10, CIFAR100 and ImageNet demonstrate a high inference accuracy with low latency. This paper proposes a simple and powerful method for the ANN-SNN conversion model, which promises in deploying the neuromorphic chips for energy-efficient inference.", "summary_of_the_review": "This paper proposes a simple and powerful method for the ANN-SNN conversion model, which promises in deploying the neuromorphic chips for energy-efficient inference.", "strength_and_weaknesses": "Strength:\nThe writing is well-written and readable in general. The overall idea makes sense and demonstrates a high performance compared with other conversion methods.\n\nWeakness:\nThe proposed approach requires the computation of additional offset spikes iteratively for each layer. Did the authors evaluate the additional computational cost of this method, e.g., the additional running time and the number of operations? Also, it would be better to provide a comprehensive comparison that incorporates this part with other conversion methods.\n\nThe key observation of using offset spikes for reducing conversion errors is derived from the QCFS function and IF model. Can it be generated to any conversion model with other activations or other neuron models?\n\nThe comparison between the computational complexity of backpropagation approaches and ANN-SNN conversion is unclear and inaccurate. What’s the exact meaning of complexity? Does the comparison incorporate the training of ANN parts for converted models?\n\nPlease provide necessary explanations for each figure in the corresponding legends, for example, the different types of bars in Fig. 3. Also, please adjust the font size in each figure to make it readable.", "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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "2oiLP8nkBId", "reviewer": "Reviewer_rBPQ", "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": "", "summary_of_the_paper": "This paper proposes the offset spike to measure the deviation of expected spikes and actual spikes of SNNs in ANN2SNN conversion, so as to avoid evaluating different conversion errors and then eliminating these errors. The authors find that the case of firing one more (or less) spike is the main reason for the degradation of SNN performance in ANN2SNN conversion. Thus they propose an optimization method to reduce the offset spike by shifting up or down the initial membrane potential. Overall they achieve SOTA accuracy on CIFAR-10/100 and ImageNet datasets with fewer time steps.", "summary_of_the_review": "This paper proposes the offset spike to eliminate ANN2SNN conversion error with a new perspective. I like the idea and would like to increase my score if the authors can solve my concerns.", "strength_and_weaknesses": "Strength:\n\n1. This paper is well-written and easy to follow.\n\n2. The idea of the offset spike is interesting. The conclusion that the case of firing one more (or less) spike is the main reason for the degradation of SNN performance in ANN2SNN conversion is impressive.\n\n3. The proposed framework is supported by theoretical analysis.\n\n4. The proposed method achieves SOTA performance in ANN2SNN conversion and comparable performance as BPTT with less computational cost.\n\nWeakness:\n1. All figures are too small to follow.\n\n2. The proposed method needs to take extra steps to acquire the remaining membrane potential, which decides the lower bound of the time steps.\n\n3. The paper lacks ablation experiments to show that the proposed shifting strategy is optimal.\n\nMore questions:\n\n1. The definition of offset spike (equation 8) lacks explanation. The authors should claim that they follow the QCFS framework with $\\lambda=\\theta$. Otherwise, it can not represent the gap between ANN and SNN output.\n\n2. It is interesting to see that the proposed method has an iterative property to reduce conversion error. However, why are the Ratio and MSE in Table 3 not zero after multiple iterations?\n\n3. Please add the ablation experiments to show that the proposed shifting strategy is optimal. The authors should compare it with other initialization strategies.\n\n4. The proposed method needs to take extra steps to acquire the remaining membrane potential. Can you comment on the implementation on Neuromorphic chips?\n\n5. I think figure 1 is under the condition of $T=L$. Am I right? How to rectify the firing rates of neurons in Figure 1c-d?\n\n6. Please clarify the value of L and T in Figure 3.", "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": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster