Adaptive Smoothing Gradient Learning for Spiking Neural Networks
Authors: Ziming Wang
Runhao Jiang
Shuang Lian
Rui Yan
Huajin Tang
Venue: Submitted to ICLR 2023 (2023)

Abstract: Spiking neural networks (SNNs) with biologically inspired spatio-temporal dynamics show higher energy efficiency on neuromorphic architectures. Error backpropagation in SNNs is prohibited by the all-or-none nature of spikes. The existing solution circumvents this problem by a relaxation on the gradient calculation using a continuous function with a constant relaxation degree, so-called surrogate gradient learning. Nevertheless, such solution introduces additional smoothness error on spiking firing which leads to the gradients being estimated inaccurately. Thus, how to adjust the relaxation degree adaptively and eliminate smoothness error progressively is crucial. Here, we propose a methodology such that training a prototype neural network will evolve into training an SNN gradually by fusing the learnable relaxation degree into the network with random spike noise. In this way, the network learns adaptively the accurate gradients of loss landscape in SNNs. The theoretical analysis further shows optimization on such a noisy network could be evolved into optimization on the embedded SNN with shared weights progressively. Moreover, we conduct extensive experiments on static images, dynamic event streams, speech, and instrumental sounds. The results show the proposed method achieves state-of-the-art performance across all the datasets with remarkable robustness on different relaxation degrees.

ADAPTIVE SMOOTHING GRADIENT LEARNING FOR SPIKING NEURAL NETWORKS
Anonymous authors Paper under double-blind review
ABSTRACT
Spiking neural networks (SNNs) with biologically inspired spatio-temporal dy- namics show higher energy efficiency on neuromorphic architectures. Error back- propagation in SNNs is prohibited by the all-or-none nature of spikes. The existing solution circumvents this problem by a relaxation on the gradient calculation us- ing a continuous function with constant relaxation degree, so-called surrogate gradient learning. Nevertheless, such solution introduces additional smoothness error on spiking firing which leads the gradients being estimated inaccurately. Thus, how to adjust adaptively the relaxation degree and eliminate smoothness error progressively is crucial. Here, we propose a methodology such that training a prototype neural network will evolve into training SNN gradually by fus- ing the learnable relaxation degree into the network with random spike noise. this way, the network learns adaptively the accurate gradients of loss landscape in SNNs. The theoretical analysis further shows optimization on such a noisy network could be evolved into optimization on the embedded SNN with shared weights progressively. Moreover, we conduct extensive experiments on static im- ages, dynamic event streams, speech, and instrumental sounds. The results show the proposed method achieves state-of- -hee--rr- performance across all the datasets with remarkable robustness on different relaxation degrees.
INTRODUCTION
Spiking Neural Networks (SNNs), composed of biologically plausible spiking neurons, present high potential for fast inference and low power consumption on neuromorphic architectures (Akopyan al., 2015: Davies et al.) 2018 Pei al., 2019). Instead of the expensive multiply-accume accumulation (MAC) operations presented in ANNs, SNNs operate with binary spikes asynchronously and offer sparse accumulation (AC) operations with lower energy costs. Additionally, existing research has revealed SNNs promise to realize machine intelligence especially sparse spatio-temporal patterss (Roy et al.) 2019). Nevertheless, such bio-mimiery with the oo-ner-none firing characteristics of spikes brings inevitably difficulties supervised learning in SNNs.
Error backpropagation the most promising methodology to develop deep neural networks. How- ever, the nonenfferet spike firing prohibits the direct application of backpropagation
SNNs. To address this challenge, two families of gradient-basse -based training methods are developed: (1) sur- rogate gradient learning (Shrestha & Orchard, 2018 Wu et al., 2018 Neftci et al. 2019) and (2) Time-based learning (Mostafa Zhang Li 2020). For surrogate gradient learning, it adopts smooth curve estimate the ill-defined derivative of the Heaviside function in SNNs. The back- propagation, in this way, could be tractable at both spatial and temporal domains in an iterative manner. Meanwhile, surrogate gradient learning could substantially benefit from the complete ecol- ogy of deep learning. It has been widely used to solve complex pattern recognization tasks (Zenke) & Vogels 2021; Neftci et al. 2019). However, the smooth curve distributes the gradient of a single spike into group of analog items in temporal neighbors ZZhan & Li 2020), which is mismatched with the inherent dynamics of spiking neurons. So we identify the problem as gradient mismatching in this paper. As result, most parameters are updated in biased manner in surrogate gradient leamn ing, which limits the performance of SNNs. Besides, different smoothness of surrogate functions may greatly affect the network performance (Hagenaars al. 2021
Li et al. 2021c).

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The time-based method is the other appealing approach. By estimating the gradients on the exact spike times, the time- based method circumvents the gradient mismatching issues in surrogate gradi- ent learning naturally. However, to obtain the exact expression of spike time, most works (Mostafal 2017 Zhang & Li, (020) suppose the firing count of each neuron remains unchanged during train- ing Yang et al. 1021) which is difficult to establish in practice. Besides, it is difficult to adapt the customized backward flows in the time-based method with auto-differential frameworks such as PyTorch, MXNet, and TensorFlow. Moreover, some special tricks are necessary avoid the phe- nomenon of dead neurons (Bohte et al. 2002). Therefore, it is not flexible to obtain deep SNNs with the time-based method.
To solve the problems, this paper proposes adaptive smoothing gradient learning (ASGL) to train SNNs directly. In general, we inject spikes as noise in ANN training and force the error surfaces of ANNs into that of SNNs. With the design of dual-mode forwarding, the smoothness factor could be incorporated into training without the need for a specific design of hyperparameter search, which could be computationally expensive. Therefore most parameters could be updated against mis- matched gradients adaptively. In addition, compared to the time-based method, ASGL backpropa- gates errors in both spatial and temporal domains without special constraints and restart mechanism.
We analyze the evolution of the noisy network with dual mode from the perspective of iterative optimization. As result, the optimization of the noisy network could be converted into minimizing the loss of the embedded SNN with the penalty of smooth factors. Experiments show the proposed method achieves sate-of-theear performance on static images, dynamic visual streams, speech, and instrumental sounds. It is worth noting that the method shows extraordinary robustness for different hyperparameter selections of smooth factors. Finally, we investigate the evolution of such hybrid network by visualizing activation similarities, network perturbation, and updated width.
RELATED WORKS
Direct Training. To eircumvent the difficulties from non-differential spikes, surrogate gradient learning approximates spike activities with pre-defined curve (Wu et al. 2019 Shrestha & Or- chard 2018 Gu al., 2019 Zenke & Vogels 202T; Fang et al.) (021b). Wu et (018)) proposed to backpropagate errors in both spatial and temporal domains to train SNNs directly with surrogate gradient. Similarly, [Shrestha & Orchard (2018) solved the temporal credit assignment problem in SNNs with the smoothness of a custom probability density function. To suppress gradient vanish- ing or explosion in deep SNNs, Zheng et al. (202I) further proposed threshold- threshol--dependent batch normalization (tdBN) and elaborated shortcut connection in standard ResNet architectures.
Gradient Mismatching. The mismatching problem in surrogate gradient learning has attracted considerable attention. [Li et al.] (2021c) optimized the shape of the surrogate gradient function with the finite difference method to compensate for this problem. Nevertheless, their method needs to ini- tialize an update step for finite difference empirically. Meanwhile, it is limited by the high computa- tional complexity of the finite difference method. Therefore, only information from the proceeding layers is adopted in the update of surrogate functions. There are still other works bypassing the dif- ficulty without using surrogate gradient. Zhang & Li 2020) handled error backpropagation across inter-neuron and intra-neuron dependenctes based the typical based scheme. Furthermore, the unifying of surrogate gradient and time-based method was suggested by (Kim et al. 20220) to fuse the gradient from both spike generation and time shift. In general, those methods are constrained by specified assumptions or coupling tricks during training. [Wunderlich & Pehle (202T) first proposed to compute the exact gradients an event-driven manner and so avoid smoothing operations by solving ODEs about adjoint state variables. Nevertheless, the approach is only verified on simple datasets with shallow networks. YYang et al. (202T) developed a novel method to backpropagate errors with neighborhood aggregation and update weights in the desired direction. However, the method based on the finite difference is computationally expensive. /eever et al. (2019) propose to sharpen the bounded ReLU activation function in ANN into the Heaviside function in SNN progres- sively. Although yielding a similar effect with ASGL, it utterly depends on hand-craft sharpening schedulers with difficulties in adaptive update considering whole network evolution. Different from the previous works, ASGL incorporates directly learnable width factors into the end-to-end training of noisy network.
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3 PRELIMINARY
Notation. We follow the conventions representing vectors and matrix with bold italic letters and bold capital letters respectively, such as and W. For matrix derivatives, we use consistent numerator layout across the paper. For function f(x:) R + Rd, we use D f(x) instead (f(h)
(x)
of
to represent the k-th derivative of f with respect to the variable æ in the absence of ambiguity. Let < M1, M2 > represent the Frobenius inner production between two matrices. For two vectors and u2, use uu Grg to represent the entrywise production. Similarly, the notation of u 02 refers to u O u for simplification. We use f o g to denote the composition of f with g.
B
Leaky Integrate-and-Fing (LIF) Model. To capture the explicit relation between input current and output spikes 8, we adopt the iterative form the LIF neuron model (Wu et al. (018)) in most experiments. At each time step t, the spiking neurons at I-th layer will integrate the postsynaptic current c'(t] and update its membrane potential u' [t]]
u [t] = >u'lt 1] (1 - s'It - 1J) + c'11
(1)
where- y = 1- 1/TT is the leaky factor that acts as constant forget gate through time. The term of (1- s - 1]) indicates the membrane potential will be reset to zero when spike is emitted the
last time step. As done in (Wu et al. 2018: Fang et al., (021b), we use simply the dot production
between weights W and spikes from the preceding layer s' "E with shift 6 model the synaptic function:
c|(4 = [[] + b² The neurons will emit spikes s'[t] whenever u'[t] crosses the threshold with enough integration of postsynaptic currents:
s'[t| = (t'(t) e -
(3)
where (zx) is the Heaviside function:
1, if 20 0
e(x) = 0, otherwise
(4)
A C-LIF variant is also applied in our experiments make fair comparison. And we provide its iterative equations in the Appendix A.2
Readout and Loss. For classification tasks, need to define the readout method matching with the supervised signal y. As done in the recent work (Li et al. 2021a Rathi et al. 2020). we fetch the average postsynaptic current in the last layer ج" N >, c where N = T7 At is the number of discrete time steps. Then the SNN prediction could be defined as the one with maximum average postsynaptic current naturally. Furthermore, we could define the cross-entropy loss removing the temporal randomness:
L( y) =- "y""loossofttaax
(5)
For simplication, we denote ğ softmax ( in the rest part the paper.
METHOD
4.1
SPIKE-BASED BACKPROPAGATION
The nature of all-or-none firing characteristics of spikes blocks the direct utilization of backpropa- gation which is the key challenge to developing spike-based backpropagation. Formally, usually
need to backpropagate the credit for the state at a specified time step 1* of 8'(4] aw in both
de"[t']
spatial domain and temporal domain (Detailed derivatives are provided in Appendix A. as follows:
2u' (1 1]
8 8 Or +
8 [t] 6'(t +
du'l1
Ou' [
Ou'it (t + 1] = y diag (1 - s'(t) - u' it o e (u (t))
(6)
(t)
+ [tt) w diag (e' (u (t))
Oul[t]
3

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Forward Function
Backward Derivative
02/2
0
a/2
-a/2
0
a/2
Figure 1: The forward and backward of the Heaviside function (green solid line) and the surrogate clipping function (purple dashed line). The red area represents the mismatching between the Heavi- side function and surrogate function considering only one spike emission.
In the above equations, the partial derivative of the Heaviside function ((x) is the Dirac-Deta func- tion which equal to zero almost everywhere. Moreover, it goes to infinity when x = 0 shown with the green line in Figure [ Such properties prohibit the backward flow in SNNs directly. Thus most researchers approximate the gradient of the Heaviside function with predefined differentiable curve (Neftci et al., 2019, Shrestha Orchard 2018). Here, we use the rectangular function (Wu eet al.) 2018), one of the most popular approximation functions with low computational complex- ity, as an example to illustrate the proposed method. The idea could be applied to other alternative approximation functions. Formally, the rectangular function could be defined as:
e' (u) = ha (u)
sign (lu
a Sl 2
a
where the width a controls the smooth degree of ha (x) and the temperature k = 1/a determines the relative steepness.
4.2 DESIGN ASGL
In surrogate learning, O' (z) is estimated by smooth function such as (z. which predicts the change rate of loss relatively larger neighborhood (Li et 2021c). However, such estimation with constant width will deviate from the correct direction progressively during the network training. not only brings difficulties to the network convergence but also affects the generalization ability with the disturbed underline neuron dynamics. Essentially, this problem comes from the mismatch-
('(z)
ing || (x) h. (1)] shown in the red part of Figure So how to adjust the width and eliminate such mismatching adaptively is an important problem for surrogate gradient learning. For ASGL, try to solve the problem without defining surrogate gradient. The method is simple but rather effective. Firstly, we derive the antiderivative function of surrogate function ha (x)):
(x)x) = [ (u,(u) du clippp 2: 0,1)
Whetstone (Severa et al. 2019) uses Ha(x) for forwarding calculation and (x for backward propagation. In this way, although there mismatching problem, it difficult to guarantee the network dynamics evolving into that of SNNs finally. In contrast, surrogate gradient learning uses ((z) and (x) for forwarding calculation and backward propagation respectively. guarantees fully spike communication but introduces the problem of gradient mismatching. Sequentially, it makes sense seek an approach combining advantages from both perspectives and training deep SNNs with matching gradients.
To implement this, the basic idea of ASGL is just couple the analog activation (a(x) and binary activation (x) through random mask m during forwarding calculation:
(x) (1 m) @ Ha( + m (()())
where m Bernoulli (p) represents independent random masking. The p controlling the propor- tion of analogs and spikes is referred to as the noise probability. To avoid the gradient mismatching from the surrogate function, we use function & detach the gradients from spikes. Mathemati- cally, & standards for the special identical mapping with the derivative 0é(x) 0. In this way, the 2x Heaviside function @(x) is taken the spike noise without error backpropagation. To guarantee
4

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Surrogate Gradient
Spiik Afferents
Post-Symaptic Currents
d
x
s(t]
Yu (
Syaapse Weights
Noise Injection
0
O(x)
Hx(x)
s[t]
<|:]
(x
Yu 1]
600 [[M]
=
Adaptive Smoothing Gradient
Figure 2: Comparasion of the computation flow between ASGL and surrogate gradient learning. The dashed line in purple area represents those operations are detached with the error backpropagation while the solid line represent the matching forward and backward flow.
the gradient still flows even when p 1, we recouple both modes and only inject the difference (ax) H (az) as noise:
(x) = H. + m ① (0(x) (a)))
(10)
The operator pipeline is visualized in Figure Surprisingly, the core idea of ASGL is simple: replace ((u((() [4]- -)) with H. (w'(1) -0) in Equation (3) during training and still adopt O [t] ()) for validation. practice, minor alterations shown Algorithm are needed compared to the surrogate gradient Icarning. Notably, the hard reset given in Equation (I) transforms into soft version at probability when the analog activations Hx(x) are propagated, despite the fact that the equations describing neuron dynamics remain unchanged. Assume H (æ> models the function from expectational membrane potential into spike probability (rate) in a short period (corresponding to one time step). The neurons have a (1 Ha (u(4)) chance of not emitting spike and maintaining the previous potential state. In the sense of expectation, the soft reset should be performed as u(() (1 (u)()) ul( rather than resetting into fixed value.
Algorithm 1 Core function in ASGL
1: Require:
The difference between membrane voltage and threshold ú (' u'(t] O: The sign T' indicates training or validation
2: Ensure: is the learnable width parameter
3: if T is true then
4: generate mask with noise probability
5: (1[ Ha (()) + m @ (((' - H. (t)
(The only line needed to update compared to the surrogate learning]
6: else
7: s' =
8:
end if
9: return s {4]
The only remaining problem is to guarantee the noisy network trained with H. could be involved into an SNN with ((x) as activation finally. Fortunately, with the perspective of mixture feedfor- ward, it could be achieved by simply setting the width as learnable (see Section 4.3 for detailed analysis):
(Ha(x) aa
[-
if Ja 30
(11)
0.
otherwise
In this way, the gradient mismatching will gradually diminish when adaptive approaches 0 against spike noise. Besides, it avoids tricky adjustments for width a which usually has a significant impact
on performance (Wu al. 2018 Hagenaars et al. 2021). The idea could also bring insights into
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surrogate gradient learning. For example, ASGL will demote surrogate gradient but still enable the adaptive width learning if we use fully spike (p = 1) for forwarding computation. Moreover, ASGL benefits naturally from the pretrained ANNs by increasing gradually p from 0. Therefore, both ANN-SNN conversion and surrogate gradient learning could be implemented in the framework of ASGI. with different settings of noise probability p.
4.3 THEORETICAL ANALYSIS
In this section, we show network dynamics of noisy networks with learnable could be evolved into that of embedded SNNs Suppose Fooise represents the noisy network used in training with (x) across all layers while Fsm denotes the target SNN embedded in Fnoise with fully spike- based calculation. The expectation over mask matrix is adopted to estimate the real loss Lnoise F, 8)
of the hybrid network Fnoise:
Ruoise 8) - Eniia (nise Eur [e(Fsnn m))]
(12)
Here, ni = m/p represents the normalized mask gathered from all layers and denotes the input spike pattern. With the perturbation analysis Appendix A.9, we have the following proposition:
Proposition 4.1 Minimizing the loss of noisy network Rnoise s) can be approximated into min- imizing the loss of the embedded SNN L (F,s) regularized by the layerwise distance between @(ū') and H Ha(
moise F, Lgn
F)
1 1-P 2p
Σ (C', diag( H. - ((ū'))
(13)
where C¹ = D2 (f. F. (G') [s'] is the second derivative of loss function l with respect to the I-th layer spike activation s' in the constructed network G which could be treated as a constant (Nagel al. 2020). G¹ denotes the network using mixed activations after 1-th layer and full spikes adopted in the front layers. To explain the proposition intuitively, we analyze the non-trivial case p # from the perspective of iterative alternate optimization. There are two steps: (1) fix weights W. optimize width a. (2) fix width optimize weights W. For the first case, as Lann [,, s) is con- stant with fixed weights, the width a tends minimize the distance between H. (ū') and O(ū'). So the penalty term diminishes and Enoise 8) approaches sm (F,s) in this step. In the second step, the noisy network with global task-related loss Enoise (F,8) 8) is optimized under a constant smooth degree. Therefore, by applying the two steps iteratively and alternately, high-performance SNN could be obtained through training noisy network even we do not increase p explicitly during training. The theoretical results have also been validated further in Fig 3b of training with fixed p. Notably, both trainable width and random noise injection with spikes are important to guarantee the first step holds on. The spike noise could be converted into the penalty on layerwise activations while the learnable a enables local optimization on it by forcing H (x: into ((x))
5 EXPERIMENTS
To validate the effectiveness of the proposed method, we conduct extensive experiments on static images and spatio-temporal patterns such as dynamic event streams, spoken digital speech, and instrumental music. Specially, we study the evolutions of network dynamics and the effect of noise rate explore whether and how injecting noise with spikes obtains the real observation about the loss landscape of target SNNs. More implementation details and energy estimation could be found in Appendix A.3 and Appendix A.oodixXA.6oee respectively.
5.1 PERFORMANCE ON STATIC IMAGES
In Table 1 we compare our work with state-of-the-arr methods the CIFAR datasets. We use the widely-used CifarNet (Wu et al. 2019) and a modified ResNet-18 structure Li et al. 2021c). As done in (Li al. 2021c Deng et al.] 20022, AutoAugment (Cubuk et al., 2018) and Cutout (De Vries & Taylor] 2017 are used for data augmentation. However, we do not adopt a pretrained ANN (Li
al. 2021c Rathi al. 2020
Rathi & Roy 2020 to initialize weights and Time Inheritance

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Table 1: Classification Performance Static Image Benchmarks.
Dataset
Method
Type
Architecture
Time steps
Accuracy($:
(Deng 2021
Conversion
400-600
90.61
STBP NediNorn Canl 9
TSSL-BP BP (hhiig q & 2020
Time-based Gradient
Surrogate Gradient
CifarNet
12
90.53
91
TL Wireere
Tandem Learning
90.98
PLIF-SNN Pang cC
Surrogate Gradient
CifarNet-B
93.50
94.74 0.10
Ours
ASGL
CifarNet
2
93.80 1: 0.11
92.84 0.21
Hyhrid Rathi et
Hyhrid
ResNet-20
92.22
CIFAR-10
STBP-MIN Zheng al.
2021
Surrogate Gradient
ResNet-19
92.34
92.92
TET Deng et al.
2022
Surrogate Gradient
ResNet-19
94.16 +: 0.03
SEW ResNet Fang
20
Surrogate Grudient
SEW ResNet- 18
91.22 0.14
94.39 + 0.18
Depike ct al
202
Surrogate Gradient
ResNet-18
93.13 = 0.07
93.66 + 0.05
95,35 0.25 95.27
Ours
ASGL
ResNet-18
94.20 = 0.01
Hyhrid Rathi et al BinarySNN Hybrid (Rarh Koy
2020
Hyhrid
VGG-11
125
67.87
&L
2020
Conversion
VGG-15
62
63,20
Hyhrid
ResNet-20
64.07
TET
Deng et al.
2022
Surrogate Gradient
ResNet-19
74.47 = 0.15
72.87 = 0.10
CIFAR-100
Dspike et 2021e
Surrogate Gradient
ResNet-18
71.68 0.12
74.59 +: 0.07
CifarNet
74.3 0.15
Ours
ASGL
74.48 :: 0.06
ResNet-18
73.19 0.04
70.73 0.07
results reproduced through publicly available
Training (TTT) (Li al., 2021c Deng al., 2022) to improve performance under low time steps. Even though, ASGL outperforms the state-of of aht surrogate gradient and conversion methods with the same or fewer time steps on both datasets. Remarkably, we also compare two special training methods TSSL (Zhang & Li 2020) and TL (Wu et al. 2021a) without the definition of surrogate gradient functions. Our method also achieves a significant remarkable on the tradeoff between accuracy and latency which indicates the effectiveness of adaptive learning employed in ASGL.
5.2 PERFORMANCE ON
PATTERNS.
Table 2: Comparasion on DVS-CII DVS-CIFAR10 dataset.
Method
Type
Architecture
Time steps
Accuracy
Streaming Rollout Kugele et al. 2020
Conversion
DenseNet
66.8
STBP-1dBN (Zheng
2021
Surrogate Gradient
Re
10
67.8
Conv3D Wu all
20
Surrogate Gradient
LIAF-Net
10
71.70
LIAF (Wi
112
Surrogate Gradient
LetF-Net
SEW ResNer (Fang cl mi
7021a
Surrogute Gradient
Wide-7B-Net
16
74.40
PLIF-SNN (Fny el al
A02TD
Surrogate Gradient
CifarNet-C C
74.80
Dspike mT. 2021e
Surrogate Gradient
ResNet-18
10
75.45
TET
Dong et a. 20/2
Surrogate Gradient
VGGSNN
10
77.40
ours
ASGL
VGGSNN
78.90
To validate that our method handles spatio-tempore error backpropagation properly. we conduct ex- periments on different datasets of spatio-temporal patterns such DVS-CIFAR10 (Li et al.) (017).
and Spiking Heidelberg Dataset (SHD) (Cramer et al.) 2020). More results of MedlyDB (Bittner ctt al. 2014) and DVS128 Gesture (Amir et 2017) could be found in Appendix A.4
Performance DVS-CIFARIO. DVS-CIFARI0 (Li et al. 2017) is challenging benchmark neu- romorphic dataset, where each sample is a record of an image of CIFAR10 scanning with repeated closed-loop motion in front of DVS. DVS-CIFARII has the same number of categories (10) and samples (Ik/class) as CI, but its recording process generates more noise, thus making classi- fication difficult. To alleviate the overfitting problem caused by data size and noise, adopt

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Table 3: Classification Performance SHD
Method
Type
#Param.
Acc.(%)
RSNN with Adaption (Yin al. 2020
Surrogate Gradient
14.13w
84.4
RSNN with Data Augmentation Faimer
2020
Surrogate Gradient
178.59w
83.2
Helerngeneuus RSNN (Perez-Niever Nieves
Surrogate Grudient
10.85w
82.7 0.8
RSNN Zenke & Vogels 202T
Surrogate Gradient
24.99w
82.0 + 2.0
Fully-comnected SNN (Pever-Nieves & Goodman
2021
Sparse Gradient
28.80w
77.5
ASGL
23.04w
86.9=
Table 4: Comparison on Image Classifi-
Table 5: Comparison on Image Reconstruction.
cation.
PSNR
SSIM
Width (a)
SG
ASGL
Width (a)
SG
ASGL
SG
ASGL
0.5
93.19
94.11
1.0
93.78
94.30
0.1
11.91
17.36
0.21
0.78
2.5
90.68
94.09
0.5
17.75
17.75
0.80
0.80
62.34
93.61
1.0
16.55
16.79
0.73
0.74
10.0
30.85
93.53
2.5
15.27
16.09
0.65
0,70
5.0
14.66
15.79
0.59
0.68
VGGSNN architecture and data augmentation method in (Deng et al., 2022). As shown in Ta- ble 2 our method achieves state-of-the-art performance (78.90%) without larger network (e.g.. ResNet- DenseNet), which outperforms existing surrogate-gradient based approaches.
Performance on Sound Datasets. The SHD dataset is spiking dataset containing 10k spoken digits generated through an encoding model simulating auditory bushy cells in the cochlear nucleus. For training and evaluation, the dataset is split into training set (8 1556 samples) and test set (2264 samples). this experiment, we train three-layer SNN (800-240-20) with recurrent synaptic connections to identify the keywords in utterances (More details about recurrent connections could be found in Appendix A.5). As shown in Table the proposed method achieves 2.5% accuracy
improvement least without any data augmentation introduced in (Cramer al.) (020 compared
to the latest results. Remarkably, we use standard LIF neurons shown in Equations (1) to (3) while
the adaptive LIF model (Yin et al. 2020) and the heterogenous LIF model (Perez-Nieves et al., 2021) are adopted to enhance the dynamics of neurons respectively.
5.3 ABLATION STUDY
In Table 4 we compare ASGL with Surrogate Gradient (SG) on CIFAR- 10 with ResNet-19 archi- tecture (Zheng et al. (021)) under N = 3 for the ablation study. The rectangular function is adopted with the same optimizer setting, seed, and weight initialization for fair comparison. Specially, we train 100 epochs with SGD optimizer and the weight decay of Se-4. The results show ASGL outperforms SG across large range of width initialization. width is selected inappropriately (a 25). In contrast, ASGL exhibits surprising robustness for dif- ferent width a. Furthermore, image reconstruction, a challenging regression task for SNNs, is also conducted to verify the effectiveness of ASGLL. Here, use h₂ (x) (tan((a2) 2 as a surrogate to show ASGL could be also applied to other functions. The fully-connected autoencoder is adopted for evaluation with the architecture of 114-128-64-32264-122----8 to-noise ratio (PSNR) and Structural Similarity (SSIM) of reconstructed MNIST images under 8 time steps. We could find the adaptive mechanism ASGL reduces sensitivity for width in SG and so shows higher performance.
catastrophic damage for SG when
Table 5 reports the peak signal-
5.4 EFFECT OF NOISE PROBABILITY
In this section, we aim to analyze how noise probability affects the performance of SNN. Firstly, we increase noise probability from 0 to 0.8 by 0.1 with every 30 epochs during the training of ResNet-19 on the CIFAR-10 dataset. As shown Figure Ba, the training accuracy of the noisy network is extremely stable while the validation accuracy of the target SNN grows erratically in the first 30 epochs. is reasonable as the noise probability is zero in the first 30 epochs and the network is purely analog without spike injections. With the noise injection of 10% spikes, the validation accuracy SNN increases rapidly around 30-th epoch at the expense of the training accuracy drop of the noisy network. That means the noisy network begins transform into target
8

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(b)
Volidaiion Actureg"
Train Accurayy
Fixed Adaptive d
0.8
0.92
0.7
0.91
0.90
0.5
Fxoe Adaptive
Validation
0.80
0.4
0.88
30
120
0.2
0.4
0.6
0.8
1.0
1.0
Epoch
Noize Probobiitty p
Noise Probobilit p
(c)
(f)
0.5
Cosine Sumilorbles
Acarod
0.8
0.6
0.5
Wilth
0.7
2.5
Vei Ifvrr
Tolt of Riledd ltd. p-
Ang. p=1.
3MG p=3
p-0.4
0.5
Veu Hyyert lie,
Tain fyyend
0.1
515,
20
100
Thin SM..
0.0
Epach
50
100
150
200
250
300
200
300
Epoch
Figure 3: (a) explores the evolution procedure by observing accuracy change with increasing spike noise. (b) and (e) examine the accuracy for different fixed selections during the test and training, respectively. (d) and (e) shows the similarities between the noisy network and embedded SNN when training with fixed p. (f) manifests the width change in the image reconstruction task.
SNNs. Interestingly, for the noise injection at the 60-th and 90-th epoch, the training accuracy is improved actually which indicates the small account of spikes in the first injection is enough for the dynamics of SNNs and the analog activations may become the obstacle for fast convergence instead. Then we explores the effect of different choices of fixed p in Figure (3p and Figure B.. Generally, low p will block the evolution into SNNs through high analog activations while excessive could not achieve the best generalization performance.
5.5 NETWORK EVOLUTION
To reveal the evolution of the noisy network, visualize accuracy changes of the noisy network and the embedded SNN with shared weights during training under p = 0.8 and = 0.5. As shown in Figure the accuracy curves of SNNs and the noisy network are extremely close in both cases. shows that the noisy network exhibits consistency in network prediction with the embedded SNN. Furthermore, we record layer-wise activations of the noisy network and the embedded SNN for each sample, and calculate the average cosine similarities S over all layers after each training epoch with ASGL. Panel (e) reports the mean and standard deviation of S across all samples in the training set of 10. According to the results, even with shared weights, the hybrid network initially has relatively low overall similarities, but after training with ASGL, the hybrid network shifts toward SNN, and the similarities increase to about 0.8. The decremental standard deviation also verifies the effectiveness of ASGL. We also evaluate the evolution of such noisy network by observing the change of learnable width (Figure BD in image reconstruction task. The width declines steadily and converges to respective values across all layers.
CONCLUSION
In this paper, we propose novel training method called ASGL to develop deep SNNs. Different from the typical surrogate gradient learning, our method circumvents the gradient mismatching problem naturally and updates weights adaptively with the random noise injection in spikes. Specifically, we train a special hybrid network with mixture of spike and analog signals where only the analog part involved in the calculation of gradients. In this way, the hybrid network will learn the optimal shapes of the activation functions against the spike noise and evolve into SNN. To validate the effectiveness and generalization of the proposed method, we analyze theoretically the evolution from hybrid networks to SNNs. Besides, conduct extensive experiments various benchmark datasets. Experimental result shows our method achieves state-of-the- performance across all the tested datasets. Meanwhile, it exhibits surprising robustness for different width selections.

---

REFERENCES
Filipp Akopyan, Jun Sawada, Andrew Cassidy, Rodrigo Alvarez-Icaza, John Arthur, Paul Merolla, Nabil Imam, Yutaka Nakamura, Pallab Datta, Gi-Joon Nam, et al. Truenorth: Design and tool flow of 65 mw million neuron programmable neurosynaptic chip. IEEE transactions on computer-aided design of integrated circuits and systems, 34310):1537-1557, 2015.
Arnon Amir, Brian Taba, David Berg, Timothy Melano, Jeffrey McKinstry, Carmelo Di Nolfo, Tapan Nayak, Alexander Andreopoulos, Guillaume Garreau, Marcela Mendoza, et al. A low power, fully event-based gesture recognition system. In CVPR, pp.
2017.
Yin Bi, Aaron Chadha, Alhabib Abbas, Eirina Bourtsoulatze, and Yiannis Andreopoulos. Graph- based spatio-temporal feature learning for neuromorphic vision sensing. IEEE Transactions on Image Processing, 29:9084-9098, 2020.
Rachel M Bittner, Justin Salamon, Mike Tierney, Matthias Mauch, Chris Cannam, and Juan Pablo Bello. Medleydb: A multitrack dataset for amnotation-intensive mir research. In ISMIR, vol- ume 14, pp. 155-160, 2014.
Sander M Bohte, Joost N Kok, and Han La Poutre. Enror-backrropagation in temporally encoded networks of spiking neurons. Neurocomputing, 48(1-4):17-37, 2002.
Tong Bu, Wei Fang, Jianhao Ding, PengLin Dai, Zhaofei Yu, and Tiejun Huang. Optimal ann- snn conversion for high-accuracy and Mitra-low-lateney spiking neural networks. In International Conference on Learning Representations, 2021.
Benjamin Cramer, Yannik Stradmann, Johannes Schemmel, and Friedemann Zenke. The heidelberg spiking data sets for the systematic evaluation of spiking neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2020.
Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc Le. Autoaugment: Learning augmentation policies from data. arXiv preprint 1xXXv:1805.09555 2018.
Mike Davies, Narayan Srinivasa, Tsung- Han Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, et al. Loihi: A neuromorphic manycore processor with on-chip learning. leee Micro, 38(1):82-99, 2018.
Shikuang Deng and Shi Gu. Optimal conversion of conventional artificial neural networks to spiking neural networks. ICLR, 2021.
Shikuang Deng. Yuhang Li, Shanghang Zhang, and Shi Gu. Temporal efficient training of spiking neural network via gradient re- re-weighting. arXiv preprint arXiv:2202. /9446, 2022.
Terrance DeVries and Graham W Taylor. Improved regularization convolutional neural networks with cutout. arXiv preprint arXiv: /708 04552, 2017.
Wei Fang, Zhaofci Yu, Yanqi Chen, Tiejun Huang, Timothée Masquelier, and Yonghong Tian. Deep residual learning in spiking neural networks. Advances in Neural Information Processing Systems, 34:21056-21069, 2021a.
Wei Fang, Zhaofei Yu, Yanqi Chen, Timothée Masquelier, Tiejun Huang, and Yonghong Tian. In- corporating learnable membrane time constant to enhance learning of spiking neural networks. In ICCV. pp. 2661-2677 2021b.
Isha Garg, Sayeed Shafayet Chowdhury, and Kaushik Roy. Det-snn: Using det distribute spa- tial information over time for learning low-latency spiking neural networks. arXiv preprint arXiv:2010.0/795. 2020.
Pengjic Gu, Rong Xiao, Gang Pan, and Huajin Tang. Stea: Spatio-temporal credit assignment with delayed feedback in deep spiking neural networks. In IJCAL, pp. 1366-1372, 2019.
Robert Gütig. Spiking neurons can discover predictive features by aggregate- learning. Science, 351(7) 2016.
10

---

Jesse Hagenaars, Federico Paredes- Vallés, and Guido De Croon. supe-supe learning of event- based optical flow with spiking neural networks. Advances in Neural Information Processing Systems, 34:7167-7179, 2021.
Song Han, Jeff Pool, John Tran, and William J. Dally. Learning both weights and connections for efficient neural networks. CoRR, (bs//506.002266, 2015.
Weihua He, Yulie Wu, Lei Deng, Guoqi Li, Haoyu Wang. Yang Tian, Wei Ding, Wenhui Wang, and Yuan Xie. Comparing snns and mns on neuromorphic vision datasets: similarities and differences. Neural Networks, 132:108-120 2020.
Jacques Kaiser, Hesham Mostafa, and Emre Neftci. Synaptic plasticity dynamics for deep continu- ous local learning (decolle). Frontiers in Neuroscience, 14:424, 2020.
Jinseok Kim, Kyungsu Kim, and Jae-Joon Kim. Unifying activation-and timing-based learning rules for spiking neural networks. NeurlPS, 33:19534-1999 19544, 2020.
Alexander Kugele, Thomas Pfeil, Michael Pfeiffer, and Elisabetta Chicca. Efficient processing of spatio-temporal data streams with spiking neural networks. Frontiers in Neuroscience, 14:439, 2020.
Souvik Kundu, Gourav Datta, Massoud Pedram, and Peter A Beerel. Spike-thrift: Towards energy- efficient deep spiking neural networks by limiting spiking activity via attention-guided compres- sion. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 3953-3962, 2021.
Michael S Lewicki. Efficient coding of natural sounds. Nature neuroscience, 5(4):356-363, 2002.
2002.
Hongmin Li, Hanchao Liu, Xiangyang Ji, Guoqi Li, and Luping Shi. Cifar10-dvs: an event-stream dataset for object classification. Frontiers in neuroscience, 11:309, 2017.
Yuhang Li, Shikuang Deng, Xin Dong, Ruihao Gong, and Shi Gu. A free lunch from ANN: towards efficient, accurate spiking neural networks calibration. ICML, volume 139, pp. 6316-6325, 2021a.
Yuhang Li, Shikuang Deng, Xin Dong, Ruihao Gong, and Shi Gu. A free lunch from ann: Towards efficient, accurate spiking neural networks calibration. In International Conference on Machine Learning, pp. 6316-6322 PMLR, 2021b.
Yuhang Li, Yufei Guo, Shanghang Zhang, Shikuang Deng. Yongging Hai, and Shi Gu. Differ- entiable spike: Rethinking gradient-descent for training spiking neural networks. NeurIPS, 34, 2021c.
Sen Lu and Abhronil Sengupta. Exploring the connection between binary and spiking neural net- works. Frontiers Neuroscience, 14:535, 2020.
Paul Merolla, John V Arthur, Rodrigo Alvarez-Icaza, Andrew S Cassidy, Jun Sawada, Filipp Akopyan, Bryan L Jackson, Nabil Imam, Chen Guo, Yutaka Nakamura, et al. A million spiking- neuron integrated circuit with a scalable communication network and interface. Science, 345 (6197):668 663,, 2014.
Hesham Mostafa.
Supervised learning based on temporal coding in spiking neural networks. IEEE transactions on neural networks and learning systems, 29(7):3227 -235,, 2017.
Markus Nagel, Rana Ali Amjad, Mart Van Baalen, Christos Louizos, and Tijmen Blankevoort. Up down? adaptive rounding for post-training quantization. In International Conference on Machine Learning, pp. 7197-7206. PMLR, 2020.
Emre O Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-basse optimization to spiking neural networks. IEEE Signal Processing Magazine, 36(6):51-6) 2019.
Jing Pei, Lei Deng, Sen Song, Mingguo Zhao, Youhui Zhang, Shuang Wu, Guanrui Wang, Zhe Zou, Zhenzhi Wu, Wei He, et al. Towards artificial general intelligence with hybrid tianjic chip architecture. Nature, 572(7767):106-111, 2019.

---

Nicolas Perez-Nieves and Dan Goodman. Sparse spiking gradient descent. Advances in Neural Information Processing Systems, 34, 2021.
Nicolas Perez-Nieves, Vincent CH Leung, Pier Luigi Dragotti, and Dan FM Goodman. Neural heterogeneity promotes robust learning. Nature communications, 12(1):1 19,, 2021.
Jordi Pons, Olga Slizovskaia, Rong Gong, Emilia Gómez, and Xavier Serra. Timbre analysis of mu- sic audio signals with convolutional neural networks. In 2017 25th European Signal Processing Conference (EUSIPCO), pp. 2744 2748. IEEE, 2017.
Nitin Rathi and Kaushik Roy. DIET-SNN: direct input encoding with leakage and threshold opti- mization in deep spiking neural networks. CoRR, abs/2008.0366 2020.
Nitin Rathi, Gopalakrishnan Srinivasan, Priyadarshini Panda, and Kaushik Roy. Enabling deep spiking neural networks with hybrid conversion and spike timing dependent backpropagation. In ICLR 2020,. OpenReview.net, 2020.
Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards spike- based machine intelligence with neuromorphic computing. Nature, 575(7784):607-611 2019.
Arash Samadi, Timothy P Lillicrap, and Douglas B Tweed. Deep learning with dynamic spiking neurons and fixed feedback weights. Neural computation, 29(3):578-66) 602, 2017.
William Severa, Craig M Vineyard, Ryan Dellana, Stephen J Verzi, and James B Aimone. Training deep neural networks for binary communication with the whetstone method. Nature Machine Intelligence, (22):86944, 2019.
Sumit Bam Shrestha and Garrick Orchard. Slayer: Spike layer error reassignment in time. arXiv preprint 1rXiv:1810.08646, 2018.
Qinyi Wang. Yexin Zhang, Junsong Yuan, and Yilong Lu. Space-time event clouds for gesture recog- nition: From rgb cameras to event cameras. In 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 1826-1833 IEEE, 2019.
Colin Wei, Sham Kakade, and Tengyu Ma. The implicit and explicit regularization effeets of dropout. In International Conference on Machine Learning, pp. 10181-10192. ICML, 2020.
Jibin Wu, Yansong Chua, Malu Zhang, Guoqi Li, Haizhou Li, and Kay Chen Tan. A tandem learning rule for effective training and rapid inference of deep spiking neural networks. IEEE Transac- tions on Neural Networks and Learning Systems, pp. 1-15, 2021a. 3095724.
doi: 10.1109/TNNLS.00.
Yujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. Spatio-tempor backpropagation for training high- gegh-performanc spiking neural networks. Frontiers in neuroscience, 12:331, 2018.
Yujic Wu, Lei Deng, Guoqi Li, Jun Zhu, Yuan Xie, and Luping Shi. Direct training for spiking neural networks: Faster, larger, better. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 1311-1318, 2019.
Zhenzhi WU, Hchui Zhang, Yihan Lin, Guoqi Li, Meng Wang. and Ye Tang. Liaf-net: Leaky inte- grate and analog fire network for lightweight and efficient spatiotemporal information processing. IEEE Transactions on Neural Networks and Learning Systems, 2021b.
Timo C Wunderlich and Christian Pehle. Event- backpropagation can compute exact gradients for spiking neural networks. Scientific Reports, 11(1):1-17, 2021.
Qu Yang, Jibin Wu, Malu Zhang, Yansong Chua, Xinchao Wang, and Haizhou Li. Training spiking neural networks with local tandem learning. NeurIPS, 2022.
Yukun Yang, Wenrui Zhang, and Peng Li. Backpropagated neighborhood aggregation for accurate training of spiking neural networks. In ICML., pp. 11852 118622 2021.
Man Yao, Huanhuan Gao. Guangshe Zhao, Dingheng Wang, Yihan Lin, Zhaoxu Yang, and Guoqi Li. Temporal-wing attention spiking neural networks for event streams classification. In ICCV, pp. 10221-10230. 2021.
12

---

Bojian Yin, Federico Corradi, and Sander M Bohté. Effective and efficient computation with timescaleeeeeeee spiking recurrent neural networks. In International Conference on Neuro- morphic Systems 2020, pp. 1-8, 2020.
Friedemann Zenke and Tim P Vogels. The remarkable robustness of surrogate gradient learning for instilling complex function in spiking neural networks. Neural Computation, 33(4):899-925, 2021.
Wenrui Zhang and Peng Li. Temporal spike sequence learning via backpropagation for deep spik- ing neural networks, In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tiees Lin (eds.), NeurIPS 2020, 2020, 2020.
Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guogi Li. Going deeper with directly-trained larger spiking neural networks. In AAAI 202/, pp. 11062-11070. AAAI Press, 2021.
A
APPENDIX
A.1
SPIKE-BASED BACKPROPAGATION
Firstly, we decompose the gradient of aggregate loss over the whole time window into different target time steps t":
ac(c' y)
deL (y 0²)
[t")
W
(14)
Oeh OW
OW
To obtain the expression of 0e we assign the credits for c²(("] into the membrane potential aW
u' [t] all time steps satisfying S 8::
[ta) aw
0e [ta] @u'lt] dct[t]
Oul(t] 0cl(t) aw!
(15)
[t" dd
Oul [1] Ow
where Be' is a three-dimensional tensor about the afferent spikes s/-) For simplification, we UW denote de*(c")
[c] as ('(t). When t < t" and I < L 1--1, 1, 5'(t) could be calculated as follows:
Ou'lt it + 1] 8 s Ors [k]
8' [t] = 6' + 11 Or' [t]
Ou [k]
Ou'l [t + 1]
7 diag (1 s'l u [t] o O''''''ii (u' (t)
(16)
Ou [1]
Ou (10
w diag (e' (n'(()))
Oul [t]
where ('(x) (0)), e' (#2), (xx))" represents the element-wise partial the colum
vector x. As for the boundary condition of the layer and time step t", we could obtain the expression of ('
8 Du4 Bu-
if L 1 and <
8'(1]=
8
[e
if anand / < L -
(17)
T.
O (2 (t))
ift t and / = L -
Then the full backward flow through time of SNNs with the LIF model could be obtained with Equations (14) (17)
13

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A.2 C-LIF MODEL
For the instrumental recognization, we adopt the current-base LIF model (C-LIF) Gütig 2016 as the basic computational unit for fair comparison. The iterative form of C-LIF could be presented
as:
[c] θ (u' [4] - 0)
e [tt] uo s's
u' a'[t] = m [t] v'(t] - e((t]
o v'[t] 3, [ 1 + c
(18)
m [t] mm |t 1] c(t
e' [t] mm m [1-1]+ 0s'
1, x20
(xx) 0, x
where uo is the normalization factor. m v' models the current integration of the synapse with the double exponential function. e'(t] simulates the refractory period in spiking neurons. The other symbols keep consistent with the standard LIF model.
A.3 EXPERIMENTAL DETAILS
Table 6: Different hyperparameters related with p.
Dataset
S
Milestones (Epochs)
CIFAR- /0/ DVS-CIFARI0 DVS125 28 Gesture
0.8
NA (300)
Tiny-ImageNer
(300)
CIFAR-100
0.6
0.8
90-th, 210-th, 270-th, 285-th (300)
SHD
0.8
0.9
30-th. 70-th, 90-th, 95-th (100)
McdlyDB
0.5
0.9
30-th, 70-th, 90-th, 95-th (100)
We use ADAM with the initial learning rate À = 0.1 for CIFAR100 and SGD with the initial learning rate of À 0.1 for CIFARIO dataset. As done (Li al., 2021e), we use AutoAugment (Cubuk et al. 2018 and Cutout Vrevs & Taylor 2017) for data augmentation in both static image datasets. Meanwhile, a cyclic cosine annealing learning rate scheduler is adopted. For the SHD dataset, we discretize the time into 250 time steps and decrease noise probability starting from 0.2. The corresponding network architecture is 700 240 20 while the neurons in the middle layer connected with recurrent synapses. For the MedlyDB dataset, we increase the noise probability at 30-th, 70-th, 90-th, 95-th epoch with the discretization of 500 time steps. In particular, we update p as 1 - (1 p) Ç at each milestone and make the ratio of analog activations attenuation at the rate of Ç. All the and C we use for each dataset are shown in Table 0 unless otherwise specified. For the results of Table we provide detailed statistics and configurations in Table
A.4 EXPERIMENTS MEDLYDB, DVS128 GESTURE, AND TINY IMAGENET
Performance on Tiny-ImageNer Tiny-ImageNet contains 200 categories and 100,000 64 x64 colored images for training, which is more challenging static image dataset than CIFAR datasets. Here, we use ha (z) 2 tanh(ar) + as the surrogate forwarding function. The initial width and decay y is set as 2.5 and 0.5 respectively. As shown in Table g ASGL still achieve competitive results compared to other methods using only 4 time steps which further verify the effectiveness of ASGL..
Performance on MedlyDB. In this experiment, explore the instruments recognization task with various music pieces in different melodies and styles. Specifically, as done (Gu et al. 2019), subtract the subset of MedlyDB which contains the monophonic stems of 10 instruments. To test our algorithm in sparse spike patterns, we adopt the efficient coding scheme based on the sparse rep- resentation theory (Lewicki 2002). Moreover,
the metric, the same network structure
14

---

Average Spike Rale
0.20
0.15
0.10
0.05
0.00
2
3
5
7
8
10
11
12
13
14
15
16
17
18
Layer Index
Figure 4: The average spike counts of each layer.
Table 7: Classification Performance DVS128 Gesture
Method
Type
Architecture
Acc. (%)
SLAYER (Shrestha & Orchard 2018
SNN( layers)
93.64
STBP DVS (He al. 2020
SNN(8 layers)
93.40
STBP-1dBN (Zeng
2021
Surrogate Gradient
SNN(ResNet17)
96.87
Temporal-wise Attention
2021
SNN(8 layers)
95.49
PLIF-SNN Fang
SNN(8 layers)
97.57
DECOLLE Kasser
Online Local Learning
SNN(4 layers)
95.54
Streaming rollouts Kugele
2020
Conversion
SNN (DenseNet)
95.56
PointNer-like ANN (Wang et al.
2019
DNN
95.32
Gradient training
RG-CNN Bi ël al. 2020
DNN
97.20
Ours
ASGL
SNN(8 layers)
97.90
(384-700-10), and the same C-LIF neuron with the previous work for a fair comparison. The results show method outperforms the others most metrics except the Recall rate of the specialized CNN (Pons et al. 2017). which designs special convolutional kernels.
Performance on DVS128 Gesture. DVS128 Gesture is a challenging neuromorphic dataset that records 11 gestures performed by 29 different participants under three lighting conditions. The dataset comprises 1,342 samples with average duration of 6.5 - 1.7 and all samples are split into a training set (1208 samples) and test set (134 samples). Considering the long sample duration and the limited sample size, follow the RCS approach (aao et al. (021 that randomly selects the starting point of the sample to maximize the use of the dataset. The time step N is set to be 60 and the network receives only one slice at each step, where the temporal resolution of each slice is adjusted to 25ms according to the tuning method in (He et al. 2020). In Table 7 our method has achieved
Table 8: Comparasion on Imageeete dataset.
Method
Type
Architecture
Time steps
Top-1 Acc.
Spike-thrift thrift (Kundu
202
Hybrid
VGG-16 16
150
51,92
DCT Garg 2020
Hybrid
VGG-13
125
56.90
SNN Calibration (Li et al. 2021b
Conversion
VGG-16
53.96
QCFS Bu 202 1
Conversion
VGG-16
53.54
Online LTI. Yang 2022
Tandem Learning
VGG-13
16
54.82
Offline Yang
Tandem Learning
VGG-13
16
55.37
Ours
ASGL.
VGG-13
4
56.57
Those results are reproduced by Yang et
2022
through publicly available codes.
15

---

the state-of-the-art performance (97.90 %) without larger network (e.g., ResNet17, DenseNet), outperforning the directly-trained approaches based on surrogate gradient. Even compared with the specially-designnn DNN approaches for neuromorphic data, our model also performs better.
A.5 RECCURENT CONNECTIONS
To enhance the memory capacity in SNNs at the temporal domain, synatic recurrence is adopted widely distinguish from the internal dynamics with decay mechanism in spiking neurons. The basic equation for such an external recurrence could be given by:
de dt
ct(t)
W
(19)
Tsyn
foodforward
recurrent
exp. decay
where the terms of decay Tsyo and Tm in Equation ①T both contribute to the internal recurrence. And the synaptic recurrence with weight v enhance the temporal memory.
A.6 ENERGY ESTIMATION
In this section, visualize the spike counts of each layer in spiking ResNet-18 shown in Figure 4 and provide the estimated energy by counting synaptic operations (SOP) compared to the ANN counterpart. Especially, the SOP with MAC presented in ANNs is constant given specified struc- ture. However, the SOP in SNN is executed by AC with lower power consumption and varies with the spike sparsity. For SNNs, the total synaptic operation with accumulation NAC is defined as:
L-IN N!
NAC ΣΣΣ
M
(20)
where fan-out f is the number of outgoing connections to the subsequent layer. N is the neuron number the /-th layer. ANNs, similar synaptic operation NMAC with more exexpensive multiply- accumulate is defined as:
N
NMA ΣΣ f
(21)
1=1
Specially, we use MAC to estimate the energy cost of the first layer as the direct current input without explicit encoding adopted in our experiments on static images. For Here, we select 1024 samples randomly and estimate the average SOP for SNNs. Meanwhile, we measure 32-bit floating-poin AC by 0.9 pJ per operation and 32-bit floating point MAC by 4.6 pJ per operation as done in (Han al,, 2015). The experimental result shows the SNN achieves 94.11% classification accuracy under two time steps on CIFAR- 10 with only 8.96% energy consumption compared to the ANN with the same architecture. We follow the convention of the neuromorphic computing community by counting the total synaptic operations to estimate the computation overhead of SNN models compared to their ANN counterparts (Merolla et al. 2014).
A.7 ABLATION STUDY ON RESNET-19 19
Table 9: Classification Performance on Music Instrument Dataset
Method
Type
Encoding Method
#Param.
Recall &
Precision( %)
Fl score( %)
Direct
Spectrogram
99.21
LSTM
Direct BP
Spectrogram
27.6w
93.31
96.08
94.62
(Samadi
DSNN (Feedback Alignment)
Spikegram
86.56
75.62
SICA Gu at 2019
DSNN (Surrogate Gradient)
Spikegram
27. 6w
97.29
97,23
97.25
DSNN (ASGL)
Spikegram
27.6w
98.58
98.52
98.59
We have added the results of ResNet- 19 (Zheng et al. 2021) under time-steps N = 3 in Table [1] (rectangular function) and Table 12 (Dspike function (Li al. (021c)). Specifically, we test the effect of different width initializations. For fairness, we use the same optimizer settings and weight initialization. All the experimental details provided in the Appendix. Notably, ResNet-19 has
16

---

Table 10: Comparsion with Surrogate Gradient Learning.
Method
SG (a 0.5)
Sa
SG (a 2.5)
SG (a 5)
SG 7.5)
(a 10)
Acc.
10.00 0.00
10.00 4: 0.00
$7.64 0.28
$1.89 0.57
66.00 4: 1.82
53.65
Method
ASCL+SG (p o-1)
(p 0.5.)
ASGL+5G(p 0.5,
ASGL(ç=0.2)
ASGL(Ç=005)
ASGL(Ç-0.8)
Acc.
10.00 + 0.00
85.72 + 0.33
53.41 #2133
89.62 0.20 89.09 + 0.17
88.83 0.09
approximate 10x operations than ResNet-18 (Li al. 2021c). Therefore we train 100 epochs for each case considering time cost. As shown in both tables, ASGI, shows robustness surprisingly on different width initializations compared to surrogate gradient with rectangular function and Dspike function. This is the main advantage of ASGL which could save the cost of hyperpa- rameter selection in practice. From Table 12, we could find that the Dspike function shows certain robustness with respect the rectangular function considering the result of = 2.5 from two tables. However, could improved further by ASGL when a I 0.5 shown in Table 12.
Experimental Setting: We train each model in ablation study for 100 epochs with initial learning rate of 0.1. We use SGD with the momentum of 0.9 across all experiments. The weight decay is set 5e-4. As done in (Li et al., (021c), (DeVries Taylor 2017) for data augmentation. Meanwhile, a cyclic cosine annealing learning rate scheduler is adopted. Besides, decay are set to 0.5. Specially, we decrease the noise rate at 20-th, 40-th, 60-th, 80-th and 95-th epoch with different Ç listed in both tables.
AutoAugment (Cubuk et al.] (2018) and Cutout
128 as batch size during training. Both threshold and
Table 11: Comparsion between ASGL and Surrogate Gradient with rectangular functions under different width initializations.
ASGL
Width
Surrogate Gradient
{ 0.2
g = 0.5
{ 0.8
a 0.5
93.19
93.89
93.93
94.11
1.0
93.78
93.83
94.30
94.15
a 2.5
90.68
93.71
94.09
93.89
a = 5.0
62.34
93.61
93.53
93.08
10.0
30.85
92.48
93.53
93.00
Table 12: Comparsion between ASGL and Surrogate Gradient with Dspike functions under different width initializations.
ASGL
Width
Surrogate Gradient
{ = 0.2
{ = 0.5
{ 0.8
0.10
82.48
90.81
91.23
89.24
a = 0.12
89.42
90.83
92.65
91.34
Q 0.14
91.41
93.20
94.03
93.36
0.5
93,93
94.02
94.34
94.28
a 1.0
93.85
93.83
94.30
93.15
Q 2.5
93.61
93.60
93.98
93.98
A.8 WIDTH UPDATR
We evaluate the evolution of such noisy network by observing the change of learnable width a. For the image reconstruction (Figure the width a declines steadily across all layers. indicates the injection of spike noise will force the noisy network to evolve into the target SNN and then optimize the target SNN in a coupled manner. Furthermore, Figure 5 shows the change a in CIFAR-10 10 classification. Interestingly, the width of the last layer in CIFARNet increases while others descend consistently. It may result from the coupling training with both goals of reducing the loss of SNN and minimizing the distance between the noisy network and SNN. Besides, it indicates that the width update should change dynamically according to different layers of the network and different training epochs.
17

---

(8)
5-
layen
layer-0
layer-1
layer
layer
Width
layer-3
Width
layer 3
layen4
layer-4
layer
layer-6
20
40
60
100
50
100
150
200
250
Epoch
Epoch
Figure 5: Fig. (a) and (b) visualize the width update in image reconstruction and image recogniza- tion, respectively.
A.9 THEORY ANALYSIS
Table 13: The symbols and corresponding definitions (explanations).
Symbol
Defination
loss function
s
input spike pattern
fl
the 1-th spiking layer with O(ū')
F
f f² o
s
the output of Fd with fully spike propagation the I-th noise spiking layer with random mask r'
g'
G'
g" gl-
g'+1
p
noise probability controlling the percent of spike mode
Proposition A.1 Minimizing the loss of noisy network Enoise (F, s) be approximated into min- imizing the loss of the embedded SNN L (F, regularized by the layerwise distance between ((u') and Ha Ha
1- Σ (C), diag (a)))) O(ū'))
Enoise (F 8) Ronn (F,
(22)
2p
Derivation. Suppose G Fd the hybrid network using spike activations O(x) in the preceding I layers and hybrid activations (a(x) after (-th layer. Morever, s' is the output spikes at I-th layer across all time steps and neurons. The detailed symbol definations and descriptions are provided in
Table 13 Then we can have:
Lasise [F, s) = E(F (s)) + E, smn m)) - L(F (s))
E(Fsmn (s)) + Em [e(G (s", m)) e(G (s"))
{(Fsnn))) + Eim
(e(Gi-1(s" (s- ,)) L(G' ()))
(23)
e(F (s)) + Σ Ey ((( ( (s)))
= E(F (8))
+
R(G
1=1
Then we adopt Taylor expansion st inspired from (Wei et al. 2020 to analyze the effect of the perturbation n at 1-th layer. From Equation (10) and Equation (3) we could obtain:
GI-1 (s'-t m) G² ((1 m)) Ha (W') mf O(a'). m)
(24)
= Gd ((1 - m') @ (H, (á') - e(ū))) + s m)
18

---

Here, we denote A (1-m m (Ha (ū') -((')) for simplication. As the expection of A is zero and [ is bounded in [0, max( L_, })1, adopt Taylor expansion around A 0 and approximate
R GG s²) here:
R(G', 8') - -n m)) (s", m)))
=Emle(G'
+A, (s), (( m))]
(25)
Die G [') + | (D² () A
Em
As A is a zero-mean vector, we discard the first-order term for expectation calculation here:
R(G', s²) En 2
A (1s)) A
(26)
Then we could take the expectation over A:
1 (D² (L Em (G)) [s] EA (AT)) 2
R
(27)
1 1-P D2 (lo Em (G)) [s] diag (Ho O(a')) 02)
2p
Here, only the diagonal elements in the covariance matrix E& independent sampling strategy presented in m. For the second order term, we could just take it as constant like (Nagel et 2020). Therefore, by substiting Equation (27 into Equation (23). we get:
[A non-zero because of the
Enoise F, 8) Re Ll ((Fam())) + 1-P Σ (C*, dia O(ū')) 02)
(28)
2p
19


Reviews:
[{"review_id": "xl2vneOiPE", "reviewer": "Reviewer_tzdq", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose a method to modify Whetstone addressing the forward-backward mismatch issue (arising from the use of surrogate gradient functions) such that the information conveyed by spikes probabilistically contributes to the forward propagation of information. This method is referred to as adaptive smoothing gradient learning (ASGL). ASGL is implemented by a simple modification of the previous computational graph so that the additional computational cost is marginal. The performance of ASGL outperforms the previous algorithms on various datasets, CIFAR-10/100, DVS-CIFAR10, SHD.", "summary_of_the_review": "The neural net considered in this work is not a spiking neural net, but it probabilistically depends on spike information as well. The activation is not in a form of binary spike, instead real-valued data. Additionally, the activation from each node is evaluated and conveyed to the postsynaptic neurons at every timestep, so that it should not be referred to as event. Therefore, the comparison of ASGL with other SNN algorithms is not fair at all. Particularly, the model is not suitable for any neuromorphic hardware, which considerably undermines the significance of the present work. ", "strength_and_weaknesses": "Strength:\n\nThe proposed method is simple and in need of marginally additional computational cost. ASGL achieves the SOTA accuracy on static datasets (CIFAR-10/100) and event datasets (DVS-CIFAR10 and SHD), which is somewhat arguable though.\n\nWeaknesses:\n\n1. The neural net for ASGL is not a spiking neural net given that the operation is not based on asynchronous events but based on real-valued outputs calculated at every timestep. Therefore, the performance comparison of ASGL with other SNN learning algorithms based on 1b spike event packets is NOT fair at all. The neural net for ASGL is rather close to DNN given that real-valued outputs forward propagate through all layers at every timestep. Further, this model is not suitable for neuromorphic hardware based on asynchronous binary event propagation. Unlike the authors’ claim that ASGL simultaneously addresses the advantages of whetstone and SNN, the model and methods rather address their disadvantages. \n\n2. For a fair comparison, ASGL should be compared with whetstone and other learning algorithms for DNNs rather than SNNs. If exist, the advantages of ASGL over DNNs should be addressed, e.g., computational/space complexities, accuracy gain, and so forth.\n", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "vFcj85nkTp", "reviewer": "Reviewer_YpMg", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 propose a new method to train spiking neural networks (SNNs) directly. The most efficient current method to do so is surrogate gradient (SG) learning where for the spike threshold the Heaviside step function is used in the forward pass, but a surrogate gradient (usually the derivative of a sigmoid function) is used in the backward pass instead of the true gradient, which causes \"gradient mismatch\". Here the authors propose that in the forward pass, one can alternate randomly between the Heaviside function and the antiderivative of the SG during training, while inference is always done with the Heaviside (otherwise it wouldn't be a SNN!). In terms of pros and cons:\n\nHeaviside:\nPros: it is the function that will be used during the inference\nCons: gradient mismatch\n\nAntiderivative of the SG:\nPros: no gradient mismatch\nCons: analog activations instead of spikes, which creates a discrepancy with inference.\n\nBy alternating between the two, the authors hope to take the best of both worlds, and they demonstrate that the idea seems to work in many experiments. \n\nThe authors also learn the width of the SG (alpha), but this has been done before.", "summary_of_the_review": "A new and promising method for direct training of spiking neural networks.\n", "strength_and_weaknesses": "STRENGTHS:\n\nAs far as I know, the alternating idea is new, and it seems to improve accuracy significantly at a marginal cost.\n\nWEAKNESSES:\n\nFig 3b is my favorite. We really see the advantage of alternating w.r.t. standard SG learning (p=1). Given that the accuracy boost is modest (~0.5%) I wonder if the authors could use error bars (with multiple seeds) to make sure that the boost is significant. Also, was this curve done with a trainable alpha? I would be curious to see if we still get a boost with a fixed alpha. Also, reporting the training accuracy would be useful (I suspect it decreases with p). Finally, the y-axis label is missing.\n\nIt's unclear what value was used for p in sections 5.1, 5.2, and 5.3.\n\nIt's unclear how the reset is done. Eq 1 shows that the potential goes to zero (hard reset) if s=1. However, with an analog s, say s=0.2, then the new potential is the previous one multiplied by gamma*0.8? This should be discussed.\n\nFang et al 2021 is cited, but it should appear in Tables 1 and 2.\n\nThe references are incomplete.\n* Eventprop (https://www.nature.com/articles/s41598-021-91786-z), which computes the exact gradients (then no gradient mismatch!) should be discussed\n* SEW ResNet (https://arxiv.org/abs/2102.04159) should be included in the tables.\n", "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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "OU99tYN2jhb", "reviewer": "Reviewer_5XQe", "summary": "", "questions": "", "limitations": "", "rating": 5, "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 describe a training method for SNNs where the binary activity is mixed with the smoothed sigmoid-like activation with a stochastic mask. The smoothed activation corresponds to the true network for which the surrogate gradient of the SNN is exact. Throughout training, the mask is released to go from a purely smoothed function at the first epochs towards a full SNN at the end. Some theoretical analysis justifies this training approach and the performance is studied empirically on many tasks.", "summary_of_the_review": "Improving surrogate gradient training is very important both for the SNN community and on the broader topic of quantized networks. The hypothesis is reasonable, but the performance does no look very good when looking carefully (see my comment above). \n\nGiven that there is no reason for this method to be specific to SNN and not to apply on binary networks, I think a good paper on the topic should also cover the background literature on straight through estimators and quantized networks which gave birth to surrogate gradient methods and have now gone deeper than SNN research in the theoretical side.", "strength_and_weaknesses": "The performance seems to improve with the suggested method but as far as I can see it is not ground breaking:\n\n1) In table 4 and table 5, I believe that we see most accurately the benefits of the training method. The fact that ASGL is robust with high width is rather trivial, since the alpha is further optimize in this case. But I find more informative that the ASGL performs 1.5% better than SG with the optimal width (alpha=1) in Table 4 and identically for image reconstruction in Table 5. This indicates that ASGL gives a meager improvement over a tuned SG which is not bad but not extremely impressive considering the additional hyperparameters that are being used.\n\nIn the other method, the method is compared with other models and seem to out perform all of them, but I understand that this probably because the authors used very strong baselines with other tricks which are independent of the ASGL method described prominently in the publication.\n\n2) The Figure 3 seems interesting to explain what is happening during training. I find the Figure 3 a) very surprising and interesting and there is perhaps something deep to understand behind that. I read the hypothetical explanation of the author for this, interesting point of view, maybe it's true. I find the other panels either trivial or hard to understand because there is no caption to explain what it means (I tried but I could not guess what is saying panel (e)).\n\nThe theory is interesting and insightful but it lacks rigor. I believe that it should not be described as a \"Theorem\". A theorem traditionally invites normally for a rigorous mathematical proof and rather as a mathematical \"analysis\" which has to be taken with some grain of salt considering the approximations which are made in the appendix. These are my main issues with this proof:\n\n4) I think a factor 1/L is missing in equation (22) in Appendix\n\n5) The Taylor expansion is performed although the Delta has no reason to be small apriori as far as I can see.\n\n6) The looks more like an approximate mathematical analysis which is perhaps insightful but the equality prominently described as a theorem in the main text is not a true equality. In my opinion, it is not ok to call that a theorem although it is insightful like so it should be presented differently in the main text.\n", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "7mPmveIqNTH", "reviewer": "Reviewer_AKTD", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper analyzes the gradient mismatching problem in direct training neural networks and proposes a novel method to solve this problem by fusing the learnable relaxation degree into the network with random spike noise and get good results on both static and events datasets.", "summary_of_the_review": "Overall, this is an interesting paper that adds a new paragraph to improve the training of spiking neural network. The originality exists but is not sufficient enough given that the problem itself was identified in another paper and not extended into a more general style in the current paper. Also, the performance examination needs to be improved to support its validity on large dataset and compatibility with other existing pipelines like the augmentation. ", "strength_and_weaknesses": "Strengths:\n1.\tIt is a novel and simple approach in the field of training spiking neural networks.\n2.\tThe presentation of this paper is generally good and easy to read.\n3.\tThe method proposed in this paper has a good performance.\n\nWeaknesses:\n1.\t1. The symbols in Section 4.3 are not very clearly explained.\n2.\tThis paper only experiments on the very small time steps (e.g.1、2) and lack of some experiments on slightly larger time steps (e.g. 4、6) to make better comparisons with other methods. I think it is necessary to analyze the impact of the time step on the method proposed in this paper.\n3.\tLack of experimental results on ImageNet to verify the method.\n\nQuestions:\n1.\tFig. 3 e. Since the preactivation values of two networks are the same membrane potentials, their output cosine similarity will be very high. Why not directly illustrate the results of the latter loss term of Eqn 13?\n2.     Is there any use of recurrent connections in the experiments in this paper? Apart from appendix A.5, I do not see the recurrent connections.\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": {}}]

Author rebuttal:

Decision: Reject