When SNN meets ANN: Error-Free ANN-to-SNN Conversion for Extreme Edge Efficiency
Authors: Gourav Datta
Zeyu Liu
James Diffenderfer
Bhavya Kailkhura
Peter Anthony Beerel
Venue: Accepted by TMLR ()

Abstract: Spiking Neural Networks (SNN) are now demonstrating comparable accuracy to convolutional neural networks (CNN), thanks to advanced ANN-to-SNN conversion techniques, all while delivering remarkable energy and latency efficiency when deployed on neuromorphic hardware. However, these conversion techniques incur a large number of time steps, and consequently, high spiking activity. In this paper, we propose a novel ANN-to-SNN conversion framework, that incurs an exponentially lower number of time steps compared to that required in the existing conversion approaches. Our framework modifies the standard integrate-and-fire (IF) neuron model used in SNNs with no change in computational complexity and shifts the bias term of each batch normalization (BN) layer in the trained ANN. To reduce spiking activity, we propose training the source ANN with a fine-grained $\ell_1$ regularizer with surrogate gradients that encourages high spike sparsity in the converted SNN. Our proposed framework thus yields lossless SNNs with low latency, low compute energy, thanks to the low time steps and high spike sparsity, and high test accuracy, for example, $75.12$% with only $4$ time steps on the ImageNet dataset. Codes will be made available. Code is available at https://github.com/godatta/SNN_meets_ANN.

Bhavya Kailkhura Published in Transactions on Machine Learning Research (04/2025)
When SNN meets ANN: Error-Free ANN-to-SNN Conversion for Extreme Edge Efficiency
Gourav Datta
gowrav douwa.datto@ccsse.edd
Deportment of Electrical, Computer, and Systems Engineering Case Western Reserve University
Zeyu Liu
luuzeyu edu
Ming Hsich Department of Electrical and Computer Engineering University of Southern California
James Diffenderfer
diffenderfer2@Hllll.ggg
Laurence Livermore Notional Laboratory
kwilkhara I @llnl.ggggg gov
Lawrence Livermore National Laboratory
Peter A. Beerel
pabeerel@use.co edu
Ming Hsieh Deportment of Electrical and Computer Engineering University of Southern California
Reviewed on OpenReview: https: // openreveew. net/ per/forum? id- NOuQKguNTO
Abstract
Spiking Neural Networks (SNN) are now demonstrating comparable accuracy to convolutional neural networks (CNN), thanks advancod ANN-to-SNN conversion techniques, all while delivering remarkable energy and latency efficiency when deployed on neuromorphic hardware. However, these conversion techniques incur large number of time steps, and high spiking activity. In this paper, propose novel ANN-to-SNN conversion framework, that incurs an exponentially lower number of time steps compared to that required in the existing conversion approaches. Our framework modifies the standard integrate-and-fire (IF) neuron model used in SNNs with no change in computational complexity and shifts the bias term of each batch normalization (BN) layer in the trained ANN. To reduce spiking activity, we propose training the source ANN with fine-grained & regularizer with surrogate gradients that encourages high spike sparsity in the converted SNN. Our proposed framework thus yields lossless SNNs with low lalency, low compule eneryy, Chanks to the low timesteps and high spike sparsity, and high test accuracy, for example, 75.12% with only 4 time steps on the ImageNet dataset. Code is available at /ttps:///ithwn wwwcchuu..ooeeeeeee cwt Hom//odatta//MN SStaa/eeNNWWsSa.sSSee ANN
1 Introduction
Spiking Neural Networks (SNNs) (Masss 1997) have emerged an attractive spatio-temporal computing paradigm for wide range of complex computer vision (CV) tasks (Pfeiffer et al., 2018 SNNs compute and communicate via binary spikes that are typically sparse and require only accumulate operations in their convolutional and lincar layers, resulting in significant compute efficiency. However, training deep SNNs has been historically challenging, because the spike activation function in standard neuron models in SNNs yields gradients that are zero almost everywhere. While there has been extensive research on backpropagation through time (BPTT) to mitigate this issue (Bellee al. 2018 Neftei et al. 2019 O'Connor al. 2018 Wu et al., 2018 Zenke and Ganguli 2018 Meng et al.) 2202aa Xiao et al.) (022). training deep SNNs from scratch is often unable to yield the same accuracies traditional iso-architecture Artificial Neural Networks (ANN).

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ANN-to-SNN conversion, which leverages the advances in state-of-the-ar (SOTA) ANN training strategies. has the potential to mitigate this accuracy concern (Sengupta et al. 2019 Rueckauer et al. 2017 Fang et al. 2021). However, since the full-precision ANN activations need to be approximated by binary spikes in the SNN layers, the number of SNN inference time steps required high. To improve the trade-off between accuracy and time sleps, previous research proposed shifting the SNN bias (Deng and Gu 2021) and initial membrane potential (Bu et al. 2022a; Hao et al. 2023a b), while leveraging quantization-awar training the ANN domain (Bu et al., 2022b Hu al. 2023a Schaefer and Joshi 2020 Sorbaro et al. 2020), inspired by the of the ANN-to-SNN conversion error incurred by the spike-driven binarization, the uneven distribution of the time of arrival of the spikes causes errors, thereby degrading the SNN accuracy. We first uncover that this unevenness error is responsible for the accuracy drop in the converted SNNs in low limesteps. To completely eliminate this unevenness as well as other errors with respect to the quantized ANN, we propose a novel conversion framework that enables exactly identical ANN and SNN activation outputs, while honoring the accumulate-only operation paradigm of SNNs. Our framework: (i) encodes both the timing information and binary value of the spikes in the membrane potential with negligible compute overhead, (ii) shifts the bias term of the BN layers in the source ANN, and (itt) modifies the IF neuron model with change in computational complexity by postponing the neuronal firings and resets after accumulation of the total input current. Our framework yields SNNs with SOTA accuracies among both ANN-to-SNN conversion and BPTT approaches with only 2-4 time steps.
estimator method (Bengio al. 2013). Although this can eliminate the component
In summary, we make the following contributions.
We analyze the key sources of error that (i) persist in SOTA ANN-to-SNN conversion approaches, and (it) degrade the SNN accuracy when using low number of time steps.
We propose a novel ANN-to-SNN conversion framework that exponentially reduces the number of time steps required for SOTA accuracy and eliminates each ANN-to-SNN conversion error. Our resulting SNN be supported in neuromorphic chips, including Loihi DDavies et al. 2018
We significantly increase the compute efficiency of SNNs by incorporating an additional loss term in our training framework, that penalizes the non-zero bits of the intermediate ANN activations, along with the task-specific loss (e.g., cross-entropy for image recognition). Further, we propose a novel surrogate gradient method to optimize this loss.
Our contributions simultancously provide lower la- tency, higher energy efficiency compared to exist- ing SNNs, especially those trained using ANN-to- SNN conversion. Additionally, our method surpasses most existing SNN training approaches in terms of
ImageNet Top-1 Accuracy
72.5
Ours
70.0
IM-Loss
OPI
performance-effccinncy trade-off, as shown in Fig. [ Note that while we use modified integrate-and- fire mechanism (postponed resets) inspired by quan- tized ANN processing, the model operates with bi- nary spikes and retains all the practical attributes of
67.5
QCFS
65.0
SurrModu
TET
62.5
60.0
SNNs (sparse spikes, accumulate operations, thresh- old checks, etc.).
25
75
100
Timesteps Spiking Activity
2 Related Works
Figure 1: Comparison of the performancc--flliincy trade-off between ourconversion & SOTA SNN training methods on ImageNet.
ANN-to-SNN conversion involves estimating the threshold value in each layer by approximating the activation value of ReLU neurons with the firing rate of spiking neurons (Cao al. 2015 Ruockauer et al. 2017 Dichl et al., 2015 Sengupta et al., 2019 Hu et al.
2018). Some conversion works estimated this threshold using heuristic approaches, such as using the maximum
(or close to) ANN preactivation value (Rathi et. al. 2020a) ). Others proposed weight normalization techniques
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while setting the threshold to unity {Kim et al., 2019 Sengupta et al. 2019
While these approaches helped SNNs achieve competitive classification accuracy on the Imagenet dataset, they required hundreds of time steps for SOTA accuracy. Consequently, there has been a plethora of research (Deng and Gu 2021 Bu
ct al. 2022b Hao et al.) 2023a b that helped reduce the conversion error while also reducing the number of time steps by an order of magnitude. All these works used trainable thresholds in the ReLU activation
function the ANN and reused the same for the SNN threshold. In particular, Dong and Gu (2021); ) Li et al.
(2021a) proposed a shift in the bias term of the convolutional layers to minimize the conversion error, with the assumption that the ANN and SNN imput activations are uniformly and identically distributed. Other
works include burst spikes (Park et al. 2019 Li and Zeng 2022), and signed neuron with memory (Wang et al. 2022a). However, they might not adhere to the bio-plausibility of spiking neurons. Some works also proposed modified ReLU activation functions in the source ANN, including StepReLU (Wang et al. 2023a and SlipReLU (Jiang et al., 2023) to reduce the conversion error. Moreover, there have been works that aim to minimize the deviation error, including Bu et al. (2022b) which proposed to initialize the membranc potential with half of the threshold value; HHa et al. (2023a which adjusts the membrane patential after observing its trend for few time steps, and Meng et al (2022b) which proposed threshold tuning and residual block restructuring Some other works explored error correction methods between ANN and SNNs, often by adapting SNNs through conversion approaches to resemble ANNs more closely (Schaefer and Joshi 2020 Sorbaro et al. 2020: Hu et al. 2023a). Lastly, some works minimized the conversion error using novel neuron models, such as inverted LIF neuron Liu et al. 2022) and signed IF neuron (Hu et al. 2023b). Our method builds upon these foundations by focusing specifically on addressing accuracy gaps at very low time steps (e.g., (4),,)) while providing substantial computational efficiency.
In contrast to ANN-to-SNN conversion, direct SNN training methods, based BPTT, aim to resolve the discontinuous and non-differentiable nature of the dinnshooding-basee activation function in the IF model. Most of these methods (Lee et al.
2016 Panda and Roy 2016 Bellec et al. 2018 Neftci et al. 2019 O'Connor
let al., 2018 Wu et al. 2018] 2021 Zenke and Ganguli i 2018 Zenke and Vogels 2021 Meng et al.) 2022a
Xiao et al. 2022 Meng et al. 2023 Guo et al.) 2022a) ) replace the spiking neuron functionality with a
differentiable model, that can approximate the real gradients (that are zero almost everywhere) with the surrogate gradients. In particular, Guo et al.] (2023a) and Guo et al. (2022b) proposed a regularizing loss and an information maximization loss respectively to adjust the membrane potential distribution in order to reduce the quantization error due to spikes. Some works optimized the BN layer in the SNN to achieve high performance. For example, Duan et al. (2022) proposed temporal effective BN, that rescales the presynaptic inputs with different weights at each time-step; Zheng et al. (2021) proposed threshold-depeeddnnt BN; Kim et al. (020) proposed batch normalization through time that decouples the BN parameters along the temporal dimension; Guo et al.] (2023b) used an additional BN layer to normalize the membrane potential. Some works extended direct SNN training to Transformers. For example, Zhou et al. (2023) introduced Spikformer using surrogate gradient learning and based self-attention, demonstrating improved energy efficiency and temporal processing capabilities. Later, Yao et al. (2023) extended this approach with Spike- Transformers, where they reformulated attention mechanisms using spike-based computations, later advancing to Spike-driven Transformer V2 (Yao et al., 2024), which introduces meta-spiking architect optimized for adaptability and next gexeratonn neuromorphic hardware. Lastly, some works have explored the use of 4 regularizers in SNN training to improve sparsity (Narduzzi et al. 2022 Ho and Chang 2021), but to the best of our knowlodge, no rescarch has specifically applied this regularization method for ANN-to-SNN conversion.
ZZhou
3 Preliminaries
3.1 ANN & SNN Neuron Models
For ANNs used in this work, a block that takes a=- input, consists of convolution (denoted by feonv), batchnorm (denoted by fBN), and nonlinear activation (denoted by fact), as shown below.
d' ffoct (F (Nooooon = fact (z')= (7² (w''' (W'a--'ma) (1)).
(1)
where W E R.oos x x denotes the convolutional layer weights (Cout and Cin are the number of output and input channels, respectively, and le is the kernel size), o² and of denote the BN running mean and variance, and
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o and 3 denote the learnable scale and bias BN parameters. All BN parameters are vectors with dimensions matching the number of output channels. Inspired by (Bu et al. 2022b), we use quantivation-cii-lloorrss (QCFS) as the activation function fact() defined as
A'' d' d'= (z))) clip
(1 2 2 2 2
.00)
(2)
where a' N xhxw Q denotes the number of quantization steps, X denotes the scalar trainable QCFS activation output threshold, and of denotes the activation input. Note that N is the batch size, and h,w are the spatial dimensions. The clip function is represented
0, i <0
clip(x, 0,
if 10<<o$p
(3)
if 2#
QCFS can enable ANN-to-SNN conversion with minimal error for arbitrary T and Q, where denotes the total number of SNN time steps.
The spike-driven dynamics of an SNN is lypically represented by the IF model where, at each time step denoted i, each neuron integrates the input current 2t) from the convolution, followed by BN layer, into its respective state, referred to as membrane potential denoted u' (t). The neuron emits spike if the membrane potential crosses a threshold value, denoted as 0¹. Assuming s (t) and s'(t) are the spike inputs and outputs respectively, 2² and of are the BN running mean and variance respectively, and 7 and B are the learnable scale and bias BN parameters, respectively, the IF model dynamics can be represented as
2(t) 2(t)=(x) (
W (t)6'-1 1 - 11) + 10)) 5t) s'()) = H( + 2'()) - 8¹),
(4)
of u'(t) (t- + (t) - s'(t)o'
(5)
where H.)) denotes the heaviside function. Note that (t), (t), and st())
xhxa where ('()) and 21()) are full-precision lensors while s()) is a binary tensor. Instead of resetting the membrane potential to zero after the spike firing, we use the reset-by-subtractin scheme where the surplus membrane potential over the firing threshold is preserved and propagated to the subsequent time step.
3.2 ANN-to-SNN Conversion
The primary goal of ANN-to-SNN conversion is to approximate the SNN spike firing rate with the multi-bit nonlinear activation output of the ANN with the other trainable parameters being copied from the ANN to the SNN. In particular, rearranging Eq. 4 to isolate the expression for ((1)) summing for t=1 to t=T, and dividing both sides by T, we obtain
Σr 8(1)0'
Et= =1(1) + (u'(0) - uT(T))
(6)
T
T
T
Then, substituting
of
(1)0'
and Z'(T) Σ 21)
((T)=
T
T
to denote the average spiking rate and presynaptic potential for the layer I respectively, obtain
d(T)= [u' (T'())-x((0)
T
Note that for very large T, (T) can be approximated with (T). Importantly, the resulting function of (((T) is equivalent to the ANN ReLU activation function, because it outputs zero for negative values of the
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Avg. deviation error
0.8
Q= VGG16 VGG16
output outputacce activation
1.00
ANN Q-8
output activation
1.00
ANN Q=8
Q-16, VGG16
SNN T=4
SNN T-2
0.75
0.75
0.6
Q-8, ResNet20
Quant. error
Quant. error
0.4
Q= 16, ResNet20
0.50
0.50
0.2
0.25
0.25
0.0
0.00
0.00
4
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50 0.75 1.00
(a)
No. of time steps
(b)
input activation
(c)
input activation
Figure 2: (a) Comparison between the average magnitude of unevenness error for different number of time steps with Q-8 and Q-16. Comparisons of the SNN and ANN output activations, o' (T) and a' respectively for (b) Q=8 and T=4, (c) Q=8 and T=2. Reducing the number of time steps from 4 to 2 increases the expected quantization error from 0.0625X to 0.125X
input (since the accumulated imput current is zero when negative) and directly reflects the positive values of the input current. This analogy is essential in understanding the transition from SNNs to ANNs using spike- based models. However, for the low in our use-case, the residual term (''(T) xx)
introduces error
in the ANN- conversion error, which previous works (Hao et al. 2023b Bu al,, 2022b) refer to unevenness error. These works also took into account two other types of conversion errors, namely quantization and clipping errors. Quantization error occurs due to the discrete nature of (T(T) which has quantization resolution (QR) of 4. Clipping error occurs due to the upper bound of (T) = 0¹, However, both these errors can be climinated with the QCFS activation function in the source ANN (sce Eq. 2) and setting 0² = X' and T=Q. This yields the same QR of T and upper bound of 0 as the ANN activation.
d'
4 Analysis of Conversion Errors
Although we can eliminate the quantization error by setting T=Q, the error increases as T is decreased significantly from Q for low-latency SNN:I This is because the absolute difference between the ANN activations and SNN average post-synaptic potentials increases as (Q-T) increases as shown in Fig bb) Note that Q cannot be too small, otherwise, the source ANN cannot be trained with high accuracy. To mitigate this concern, we propose to improve the SNN capacity low T by embedding the information of both the timing and the binary value of spikes in each membrane potential. As shown later in Section 5 this climinates the quantization error at log Q. This results in an exponential drop in the number of time sleps compared to prior works that require T=Q (Bu et al., 2022b). As our work already enables a small value of T, the drop in SNN performance with further lower T Q becomes negligible compared to prior works. Moreover, at low timesteps, the unevenness error increases as shown in Fig, 2 aa, and even dominates the total error shown in Fig. Right), which highlights its importance for our use case. Previous works (Hao et al. 2023a b) attempted reduce this error by observing and shifting the membrane potential after some number of time steps, which dictates the upper bound of the total latency. Morcover, Hao ct al. (2023a requires iterative potential correction by injecting or eliminating one spike per neuron at time, which also increases the inference latency. That said, the unevenness error is difficult to overcome with the current IF modols. To climinate the unevenness error, (T) must fall in the range [0, 8') (uu et al. 2022b However, this cannot be guaranteed without the prior information the post-synaptic polentials (up to T time steps). The key reason this cannot be guaranteed is the neuron reset mechanism, which dynamically lowers the post-synaptic potential value based on the input spikes. By shifting all neuron resets the last time step T, and matching the ANN activation and SNN post-synaptic values at cach time step, completely eliminale this unevenness error, as shown in Section 5
Note that T cannot always be equal to for practical purposes, since may want multiple SNNs with different number of steps from a single pre-trained ANN
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5 Proposed Method
In this section, propose our ANN-to-SNN conversion framework, which involves training the source ANN using the QCFS activation function (Bu et al. 2022b and 1) bit-wise fine-grained & regularizer, followed by 2) shifting the bias term of the BN layers, and 3) modifying the IF model where the neuron spiking mechanism and reset are pushed after the current accumulation over all the time steps.
5.1 ANN-to-SNN Conversion
To enable lossless ANN-to-SNN conversion, the IF layer output should be equal to the bit-wise representation of the output of the corresponding QCFS layer in the Md block, which can be represented as (t)= Vt E [1. 71, where of denotes the tth bit of starting from the most significant bit. This ensures that the cumulative spike train over 2 time steps reconstructs the full quantized activation value of the ANN.
We first show how this is guaranteed for the input block and then for any hidden block by induction.
Input Block: Similar to prior works targeting low-latency SNNs (Bu et al., 2022b a Rathi et al. 2020b): we directly use multi-bit inpuls that incur multiplications in the first layer, whose overhead is neg regligible in a deep SNN. Hence, the input to the first IF layer in the SNN (output of the first convolution, followed by BN layer) is identical to the first QCFS layer in the ANN. The first QCFS layer yields the output a² with log, Q bits. The first IF layer also yiclds identical outputs s (t) = a at the tth time step, with the proposed neuron model as shown later in Eqs. 9 and T0
To incorporate the information of both the firing time and binary value of the spikes, we multiply the input (t) of the IF layer (i.c., output of the convolution followed by a BN layer) in the 1th block by 20t-1) at the gth time step, which can be easily implemented by left shifter. Note that the additional compute overhead due to the shifting is negligible as shown later in Section 6.3 The resulting SNN The input of the corresponding ANN QCFS layer is ,BN (BN( (o'n)(d where a! can be denoted as Σ 10-gg/1(( by induction.
Hidden Block:
input current in the t block is computed as s²(1) = JBN (2
Condition J: For lossless conversion, let us first satisly that the accurullated input current over T time steps is equal to the input of the corresponding QCFS layer in the 1th block.
Mathematically, representing the composite function f BNN (frone (-) as g ANN and gSNN for the source ANN and its converted SNN respectively, Condition can be re-written as
Σ g (N
1 (1)= ANN
Σ k ( (1)
(8)
where k= However, this additive property does not hold for any arbitrary source ANN and its converted SNN, duc to the BN layer. We satisfy this property by modifying the bias of each BN layer during ANN-to-SNN conversion, as shown Proposition I below, whose proof in Appendix A.2.
Proposition I: For the t block in the source ANN, let us denote W as the weights of the convolutional layer, and μ², o², 7², and 8 as the trainable paramcters of the BN layer. Let denote the same parameters of the converted SNN for as W Hls de Yes and B.. Then, Eq. holds true if W W pr = pé, of o', 7t =
and 3. g' T + (1 +) 2
et
Proposition II: If Condition I (Eq. is satisfied and the post-synaptic potential accumulation, neuron firing, and reset model adhere to Eqs. 9 and 10 below, the lossless conversion objective i.c., s' s(t)=a! Vt E [1,T] is satisfied for any hidden block I.
of 16t++ +
(t) = %
(9)
T
3 (t), (t)= (n'(t) (u(()) 0²) 1) 1)= a d'() 5((1) d
n/ a1(1==)
Σ
(10)
2t
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ANN trained using QCFS activation
0.16
76
Come
BN
GL''S
Comw
BN
OCRS
0.14-
Quartization
Clipping
74
Unevernees
54. of unawerness
New neuron, that segregates accumulation emiesion phases, Adds the BN biss shift the initial membrane patential
Broor Values
0.10
Single ANN block
88
Compute BN
0.08
66
Conv
DC7S
shift
0.06
0.04
Converted SNN
0.02
60
Comv
New Neuron
ON
New Neuron
Comv
BN
New F Neuron
0.00
-9
Number of time steps
Figure 3: (Left) Proposed A.NN-to-SNN conversion framework, encompassing i) training of the source ANN using the QCFS activation function, ii) computing the shift of the bias term of the BN layers, and copying the other trainable parameters and iii) modification of the TF neuron. (Right) Comparison of the average magnitude of quantization, clipping, and unvveneess errors between the ANN and SNN.
The proof of Proposition II is shown in Appendix A.2. Our conversion framework is illustrated in Fig. Leftt). Note that our neuron model postpones the firing and reset mechanism until after the input current. is accumulated from the incoming spikes emitted over all the time steps in the previous layer. Hence, our model does not change the computational complexity of the traditional IF model. Moreover, our neuron model can be supported in programmable neuromorphic chips, that implements current accumulation, threshold comparison, and potential reset. independently in modular fashion. Since our model needs to acquire a (T),, before transmitting the spikes at any time step to the subsequent layer, it requires layer-by-layer propagation, as used in advancod conversion works (Hao et al.] 2023a b). However, this does not prohibit the asynchronous computations that can be accelerated by an asynchronous accelerator such as Loihi. In particular, spikes are transmatted to the next. layer as soon as they are computed. Moreover, our implemented framework adheres to this scheme and thus our reported accuracies are consistent with the asynchronous implementation. The only constraint the layer-by-laeer propagation incurs is that all time steps of the previous layer must be computed before the spikes of the first time step of the next layer can be computed. However, this constraint does not impose any penalty, as layer-by-layer propagation is superior compared to its alternative by-tpp propagation in terms of system efficiency as shown in Appendix A.3.
While our approach of separating the aggregation and emission phases similar to Liu et al. (2022) there are notable differencce that result improved SNN accuracy, particularly at low time steps. Firstly, our method embeds both the timing and binary value of spikes within the accumulated input current (as indicated by the term 2t in Eq. 9). Secondly, model completely climinates the conversion error, in contrast, Liu et al. (2022 empirically shows that their inverted LIF model only reduces (not eliminate) the conversion error.
provide mathematical proof demonstrating that our proposed neuron
5.2 Activation Sparsity
Although our proposed framework can significantly reduce T while eliminating the conversion error, the spiking activity does not reduce proportionally. Note that spiking activity is defined as the average number of spikes per neuron in the entire SNN over all the time steps i.e., the total inferencing window. In fact, we can see from Fig. 7(a) that the spiking activity of a VGG-16 based SNN evaluated on CIFAR10 drops only ~% (36.2% to 33.0%) when T decreases from 8 to 4. We hypothesize this is because the SNN tries to pack a similar number of spikes within the few time steps available. To mitigate this concern, we propose a fine- the-grained regularization method that encourages more zeros in the bit-wise representation of the source ANN. As our approach enforees similarity between the SNN spiking and ANN bit-wise output, this encourages more spike sparsity under low T, which in turn, decreases the compute complexity of the SNN when deployed neuromorphic hardware. The training loss function (Lrotet) of our proposed approach is shown below in Eq. 11 where at i.! denotes the tth bit of the 4t activation value in layer I, LCE denotes the moss-entrops loss caleulated the softmax output of the last layer L,, Lsp denotes the proposed e, regularizer loss, and À is
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the regularization coefficient.
Lootal=LoE + NLsp-LoE +X
(11)
Note that we only accumulate (and do not spike) the post-synaptic potential the last layer I, and hence, we do not incorporate the loss due to id for I=L. Since a, {0, 1}, its gradients are either zero or undefined, and so, we cannot directly optimize Lsp using backpropagation. To mitigate this issue, inspired by the straight-through estimator (Bengio al., 2013), we propose a form of surrogate gradient descent as shown below, where a denotes the t-bit activation of neuron i in layer l:
ÖLsr da
L
N
T
dug"
day' Da
[1, ir0< X
(12)
Da
0. otherwise
6 Experimental Results
In this section, we demonstratc the efficacy of our framework on image rocognition tasks with CIFAR-10 (Leeun al., 1998), CIFAR100 (Krizhevsky, 2009), and ImageNet datasets (Deng et al., 2009). Similar prior works, we evaluate our framework VGG-16 (Simonyan and Zisserman 2014), ResNet18 (He et al.j 2016), ResNet20, and ResNet34 architectures for the source ANNs. To the best of our knowledge, are the
first to yield low latency SNNs based on the MobileNetV2 (Sandler et al., 2018) architecture. We compare our method with the SOTA ANN-Lo-SNN conversion methods including Rate Norm Layer (RNL) DDing )021), Signed Neuron with Memory (SNM) (Wang et al., 2022a), radix encoded SNN (radix-SNN)
ct al.
Wang et al.
2022b), SNN Conversion with Advanced Pipeline (SNNC-AP) (Li et al. 2021a), Optimized
Potential Initialization (OPI) (Bu al. 2022a),. QCFS (Bu et al.)
2022b). Bridging Offset Spikes (BOS) (Hao
et al.
2023a
Residual Membrane Potential (SRP) (Hao et al. (023b) and direct training methods including
Dual Phase
Wang et al. 2023b Diet-SNN (Rathi et al.) 2020b), Information loss minimization (IM-Loss)
Guo el al.
Differentiable Spike Representation (DSR) (Li et al.] 2021b), Temporal Efficient Training (PLIF) (Fang et al. 2021), ; RecDis-SNN (Guo et al., Membranc Potential Reset (MPR) (Guo et al.) 2022c Temporal Effective Batch Normalization
2022b1,
Deng et al.
2022
parametric leaky-integrate-and
2022ah,
(TEBN) (Duan el al. 2022), and Surrogate Module Learning (SML) (Deng el al., 2023 More details about.
the proposed conversion algorithm and training configurations are in Appendix A.1.
6.1 Efficacy of Proposed Method
To verify the efficacy of our proposed method, we compare the accuracies ob- tained by our source ANN and the verted SNN on CIFAR datasets. As shown in Fig. 4 for both VGG and
80
Test Test Accuracy (96)
78
94
76
74
92
70
ResNet architectures, the accuracies ob- tained by our source ANN and con- verted SNN are identical for T=log2Q. This is expected since we ensure that both the ANN and SNN produce the same activation outputs with the shift of the bias term of cach BN layer. Hence, unlike previous works, there is no laser-wise error that gets accumu- lated and transmitted to the output layer. However, the SNN test accuracy starts redueing for lower T, which is due the difference between the ANN and SNN activation outputs, but is still higher than existing works at the same T as shown below.
(a)
ResNer18
VCG16
Test
66
(b)
VGG16
ResNec18
RasNer12
ResMer20
No. of time steps
No. of time steps
Figure 4: Comparison of the test accuracy of our conversion method for different time steps with Q = 16 on (a) CIFAR10 and (b) CI- FAR100 datasets. For T=log=:=44 the ANN & SNN test accuracies identical. The source ANN accuracies shown in dotted lines.
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Architecture
Method
ANN
T=2
T=4
T=6
T=8
T=16
7=32
RNL 92.82%
57.90%
85.40%
SNNC-AP
95.72%
93.71%
OPI 94.57%
90.96%
93.38%
94.20%
VGG16
BOS" 95.51%
95.36%
95.46%
95.54%
95.61%
Radix-SNN
93. 93.84%
94.82%
QCFS
95.52%
91.18%
93.96%
94. 74%
94.95%
95.40%
95.54%
Ours 95.82%
94.21%
95.82%
95.79%
95.82%
95.84%
95.81%
OPT 96.04%
66.24%
87 87.22%
91.88%
BOS" 95.64%
95.25%
95. 95.45%
95.68%
95.68%
ResNet18
Radix-SNN
94.43%
95.26%
QCFS
95.64%
91.75%
93.83%
94.79%
95.04%
95.56%
95.67%
Ours 96.68%
96.12%
96.68%
96.65%
96.67%
96.73%
96.70%
OPI 92.74%
66.24%
87 87.22%
91.88%
BOS 91.77%
89.88%
91.26%
92.15%
92.18%
ResNet20
QCFS
91.77%
73.20%
83.75%
83.79%
89.55%
91.62%
92.24%
Ours
93.60%
86.9%
93.60%
93.57%
93. 33.66%
93.75%
83.82%
Table 1: Comparison of our proposed method to existing ANN-to-SNN conversion approaches on CIFAR10.
Q = 16 for all architectures, x-1e-8. *BOS incurs at least 4 additional time steps to initialize the membrane potential, so their results are reported from T>4.
Architecture
Method
ANN
T=2
T=1
T=6
T=8
T=16
T=32
SNM
73.18%
64.78%
SNNC-AP
75.36%
63.64%
OPI 93.63%
60.30%
ResNet34
BOS"
71.22%
67.12%
68.86%
74.17%
73.95%
SRP"
74.32%
57.22%
67.62%
68.18%
Radix-SNN
72.52%
73.45%
73.65%
QCFS
74.32%
35.06%
59.35%
69.37%
Ours 75.12%
54.27%
75.12%
75.00%
75.02%
75.10%
75.14%
SNNC-AP
73.40%
37.43%
MobileNetV2
QCFS
69.02%
0.20%
0.26%
0.53%
1.12%
21.74%
58.45%
Ours
69.02%
22.62
.2%
68.81%
68.89%
68.98%
69.02%
69.01%
Table 2: Comparison of our proposed method to existing conversion methods on ImageNet. Q=16 for both ResNet34 and MobileNetV2, and >=5e-10. "OSS and SRP incurs at least 4 and 8 additional time steps to initialize the potential, so their results are reported from T>4 and 7>8 respectively.
6.2 Comparison with SOTA
We compare our proposed framework with the SOTA ANN-to-SNN conversion approaches on CIFAR1O and ImageNet in Table [ and 2 respectively. For low number of time steps, especially T54, the test accuracy of the SNNs trained with method surpasses all the existing methods. Note that our accuracy slight drops at T=6 likely because the source QCFS ANN being trained with Q=16 (equivalent to 4-bit precision), leading to optimal performance at T-4. However, this drop (~0.1%) is within expected noise levels. Our SNNs can also outperform some of the recently proposed SNNs that incur even higher number of time steps. For example, QCFS reported a test accuracy of 94.95% at T==; our method can surpass that accuracy (yield 95.82%) at T=4. Note that Hao et al. (2023b requires additional time steps to capture the temporal trend of the membrane potential. The authors reported 4 extra time steps for the accuracy numbers that are shown in Table T As result, they require at least 5 time steps during inference, and their reported accuracies
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are lower compared to our SNNs at iso-time-step across different architectures and datascts. Moreover, our approach results in 22% increase in test accuracy on both CIFARIO and ImageNet compared to radix encoding (Wang et al. 2022b), that proposed a shifting method similar to our left-shift approach, for low time steps (<4). This demonstrates the efficacy of our BN bias shift and ncuron model. Morcover, as shown in Table 3 our low-latency accuracies are also higher compared other SOTA yet memory-expensive SNN training techniques, such as BPTT and hybrid training, at iso-time-step. Lastly, compared to these, our conversion approach leverages standard ANN training with QCFS activation and requires changing only one parameter of each BN layer, that is not repeated across time steps, before the SNN inference process.
6.3 Energy Efficiency
Our modified IF model incurs the same number of membrane potential update, neuron firing, and reset, compared to the traditional IF model with identical spike sparsity. The only additional overhead is the left shift operation that is performed on each convolutional layer output in each time step. As shown in Table G in Appendix A.4, a left shift operation consumes similar energy as an addition operation with identical bit- tit-prrcision. However, the total number of left shift operations is significantly lower than the number of addition operations incurred an SNN for the spiking convolution operation. Intuitively, this is because the computational complexity of the spiking convolution operation and the left shift operation are Cinaa and (Cout W') respectively, where s denotes the sparsity. Note that k denotes the kernel size, Can and Cout denote the number of input and output channels respectively, and H and W denote the spatial dimensions of the activation map. Even with sparsity of 90%, for =51 and k=3, in ResNet18, we have Ha compute energy, which is the the energy incurred by the floating point operations, specifically accumulate operations in SNNs (Rathi and Roy 2023 Datta et al. 2024), across both VGG and ResNet architectures. Moreover, left shifts can also be supported in programmable neuromorphic chips.
Cou 406.8. Hence, as shown in Fig. faa,, the left shifts ineur negligible overhead in the total
Our SNNs enjoy superior energy ef- ficiency compared to existing SNNs with a higher number of time steps, our method effectively reduces the quired time steps while preserving ac- curacy. By maintaining the same com- putational complexity as a quantized ANN and incorporating modifications to the IF model, our approach enables a more energy-efficient SNN. Moreover, our fine-grained regularizer significantly reduces the spiking activity of the net- work. As shown in Fig. 5[))((), with VGG16, we can obtain& a 1.64x reduc- tion for CIFAR10 and 2.40x reduction for CIFAR100. For 18 on CIFAR10 and ResNet-34 on CIFARI00, the reduction factors are 2.41 x and 2.33x respectively. Compared SOTA conversion approaches (Bu al., 2022b Hao al. 2023a). we obtain 3.73-10.760 70x reduction in spiking activity. This reduced spiking activity linearly reduces the compute energy. Additionally, our low-latency SNNs significantly reduce the memory access cost, which is dominated by the successive read and wrile operations of the membrane potentials in each time step. Thus, our proposed low-latency conversion framework, coupled with high spike sparsity, can significantly reduce the combined system energy. Detailed energy comparisons with ANNs and additional analysis are in Appendix A.4.
Comparator
Reset
Our (k=1e-E)
Oun (KHO)
GCTS
BCS
1.0
(a)
selliing activity
(b)
[c]
0.6
1.0
0.4
0.2
0.5
0.0
VGG16
ResNet20
VG616
Res
VGGLE
RenNie20
Figure 5: (a) Comparison of the compute energy of each SNN operation with X=le-8 on CIFAR10. Comparison of the spiking activiles of the SNNs obtained via our and SOTA conversion methods on (b) CIFAR10 and (c) CIFAR100 with VGG16 and ResNet20. In (a), LS denotes the left shift operation, and CE denotes compute energy.
6.4 Ablation study of Neuron Model
We conduct ablation studies of our proposed encoding and conversion framework using the traditional IF model. As shown in Table 4, the SNN accuracy drops compared to the ANN counterpart, and the degradation is severe for low (2-4) time steps. This is due the deviation error that appears with the normal IF model, and increases significantly at low time steps, dominating the total error. These results validate our hypot
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Dataset
Method
Approach
Architecture
Accuracy
Time Steps
Dual-Phase
Hybrid
R8sNet18
93.27
IM-Loss
BPTT
ResNet19
95.40
MPR
BPTT
ResNet19
96.27
TET
BPTT
ResNet19
94.44
CIFARIO
RecDis-SNN
BPTT
ResNet
95.53
TEBN
BPTT
ResNet19
95.58
SurrModu
BPTT
ResNet
96.04
Ours
ANN-to-NN
ResNet18
96.68
Dspike
Supervised learning
VGG16
71.24
5
Diet-SNN
Hybrid
VGG16
69.00
5
SEW ResNet
BPTT
ResNet34
67
4
ImageNet
IM-Loss
BPTT
VGG16
70.65
5
RMP-Loss
BPTT
ResNet34
65
4
SurrModu
BPTT
ResNet34
68.23
4
SDT V2
BPTT
Mela-Spikeforner
80.00
4
Spikformer V2
BPTT
Spikformer V2-8-512
80.38
4
Ours
ANN-1o-SNN
ResNet34
75.12
4
Table 3: Comparison our method with SOTA BPTT and hybrid training approaches.
Architecture
Left shift
BN bias shift
Modified
T-2
T-4
T-
T-
T'=16
x
X
91.08%
93.82%
94.68%
94.90%
95.33%
X
x
92.42%
94.80%
95.17%
95.28%
95.21%
VGG16
93.03%
95.12%
95.24%
95.18%
95.21%
v
93.33%
95.23%
95.45%
95.45%
95.32%
V
94.21 %
95.82%
79%
95.82%
95.84%
71.42%
83.91%
84.12%
88.72%
92.64%
76.21% %
90. 18%
91.92%
92.49%
92.62%
ResNet20
X
x
76.10%
91.22%
91.43%
92.40%
92.62%
V
79.86%
91.81%
92.07%
93.24%
93. 93.48%
86.92%
93. 93.60%
93.57%
93.66%
93.75%
Table 4: Ablation study of the different components of our proposed method on CIFAR10 with VGG16 and ResNet20.
presented in Section 3. Additionally, when we use the normal IF model, the encoding and bias shift of the BN layers still yield noticeable accuracy increase compared to the QCFS training method that our work is based on, especially for 2-4 time steps. For hardware that can only support the standard IF model, our conversion framework employing this model yields superior accuracy compared to most of the existing SNN works, shown in Table 1.
6.5 Comparison with quantized ANN
While SNNs were originally proposed to mimic the neural mechanism of humans, SNNs also yield high energy efficiency arising from the spike sparsity and accumulate-only operations, while maintaining statc-of-the-art accuracy. Our method enhances the energy efficiency compared to cxisting SNNs by drawing inspiration from aclivation quantized ANNs and proposing a new neuron model and batch norm (BN) bias modification strategy, that ensures the ANN and average SNN outputs are identical at each layer. While this implics some degree of similarity with quantized ANNs, marrying the efficiency benefits from the quantization in ANNs and sparsity in SNNs helps enable low-power and low-latency neural networks, particularly given the rise of neuromorphic chips.
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Architecture
T=Q
SNN Acc. (%)
QANN Acc. (%)
2
94.21
94.73
VGG16
3
95.30
95.37
4
95.82
96.02
2
86.90
86.73
ResNet220
90.77
91.22
4
93.60
94.06
Table 5: Comparison of accuracy of the SNNs obtained via our conversion framework with quantized ANNs (QANN) on CIFARIO.
As shown in Table 5 with VGG16 and ResNet20 on CIFARIO, our SNNs incur only a marginal reduction of test, accuracy compared to quantized ANNs. This reduction due our fine-grained & regularizer that trades accuracy for spiking activity. Note that for a fair comparison, we use T = Q, where T is the total number of SNN time steps, and Q is the activation bit-width of the ANN. While our SNNs incur slight drop in accuracy, they are significantly more energy elficient. than quantized ANNs. First, quantized ANN accelerators do not typically leverage activation sparsity that avoid computation when any of the bits in the activation are zero. Secondly, they require quantized multiply-and-accuumlla (MAC) operations, which incur significantly more energy compared to accumulate (AC) operations required by SNNs. For example, a 4-bit integer MAC operation incurs 2.3x higher compute energy compared to a 4-bit. integer AC operation mm CMOS technology, as observed in our in-house FPGA simulations. Thirdly, our SNNs provide additional spike sparsity (on top of the natural spike sparsity) due to fine-grained regularizer, which further increases the energy-efficieney As result, SNNs incur ~5.1 lower compute energy for T Q 4 as shown in Table 11 in Appendix, when averaged over VGG and ResNet architoctures, on CIFAR10 and ImageNet.
7 Conclusion
In this paper, we first uncover the key sources of error in ANN-to-SNN conversion that have not been completely eliminated in existing works. We propose a novel conversion framework, that introduces modified IF neuron model and shifts the bias term of each BN layer of the source ANN, before the SNN inference. Our neuron model has identical compute and memory complexity compared to the traditional IF neuron model. Our framework completely eliminates all sources of conversion errors when use the same number of time steps as the bit precision of the source ANN, We also propose a fine-grained l regularizer during the source ANN training that minimizes the number of spikes in the converted SNN. This significantly increases the compute efficiency, while the ultra-low latency increases the memory efficiency of our SNNs. To the best of our knowledge, our work is the first. achieve ultra-low latency and compute energy, while still achieving the SOTA test accuracy on complex image recognition tasks with SNNs.
8 Acknowledgements
Gourav Datta, Zeyu Liu, and Peter A. Beerel gratefully acknowledge gift funding from Intel Labs. James Diffenderfer and Bhavya Kailkhura conducted this research under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE- DE-AC52-07NC supported by the LINL-LDRD Program under Project No. 22-DR-009 (ILNI-JRNT--57885))
A2224477 and was
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A Appendix
A.1 Network Configurations and Hyperparameters
We train our source ANNs with average-pooling layers instead of max-pooling as used in prior conversion works
(Hao et al. 2023a Bu et al. 2022b) We also replace the ReLU activation function in the ANN with
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QCES function as shown in Eq. 2, copy the weights from the source ANN to the target SNN and set the QCFS activation threshold X' equal to the SNN threshold 0². Note that X is a scalar term for the entire layer to minimize the compute associated with the left-shiff of the threshold in the SNN, We set the number of quantization steps Q to 16 for all networks on all datasots.
We leverage the Stochastic Gradient Descent optimizer (Bottou 2012) with a momentum value of 0.9. We use an initial learning rate of 0.02 for CIFAR-10 and CIFAR-100, and 0.1 for ImageNet, with a cosine decay scheduler (Loshchilov and Hutter 2017 to lower the learning rate. For CIFAR datasets, we set the value of weight decay to 5x10-*, while for ImageNet, it is set to 1x10- Additionally, we leverage advanced input augmentation techniques to boost the performance of our source ANN models (DeVries and Taylor 2017 Cubuk et al., 2019), ; which can eventually improve the performance of our SNNs. The models for CIFAR datasets are trained for 600 epochs, while those for ImageNet are trained for 300 epochs. All experiments are performed on an NVIDIA V100 GPU with 16 GB memory.
A.2 Proof of Propositions & Statements
Proposition-I: For the 1th block in the source ANN, let us denote W as the weights of the h hidden convolutional layer, and μ², o', x', and I as the trainable parameters of the BN layer. Let us denote the same parameters of the converted SNN for as Wt pho oe Yo and 3 Then, Eq. holds true if W = W
p._ = d. = d', %e = 7, and 8% = _ + (1 3)
Proof: Substituting the value of SNN for the SNN in the left-hand side (LHS) which is equal to the accumulated input current over T time steps, Et= 21, and ANN in the right-hand side (RHS) of Equation we obtain
E (%6 2 11 11)0011 v/
+ t1. = 7' - (2 (1)) + B¹)
o²
W of
Et 1 20-1 (1) + T Mhad s ΣtT -= ol (1) + (3) روي')
of If we assert % ,,, W! = =W', of = =o,, the first terms of both LHS and RHS are equal. Substituting 10 = =n", W WI, and of af with this assertion, LHS-RHS if their second terms are equal, i.c T( 1 (8' =ج/
T - 8 (T 1
B. = # + (1 4)
Proposition-14 II: If Condition I (Eg. is satisfied and the post-synaptic potential accumulation, neuron firing, and reset model adhere to Eqs. for any hidden block I.
9 andd 10. the lossless conversion objective i.e., s(()=aa Vt E [1,T] is satisfied
In Eqs. and 10 -11)= - Et, (t) is the original LHS of Eq. g Given that Eq. is satisfied due Proposition-l, we can write (1) h², where ht' is the input to the QCFS activation function of the 1th block of the ANN. The output of the QCFS function is denoted as a' = fact (h'), whose 4th bit starting from the most significant bit (MSB) is represented as af. We can check if af is zero one, iteratively starting from the MSB, using binary decision tree approach where we progressively discard one-half of the search range for the subsoquent bit checking. With the maximum value of h' being X, and = 8' (see Section 3.2), af = (h' )) (1)(1)) 2) s'))- To compute al, we can lower h by half of the previous range, by first updating he' as h/ = - of 2. and then calculating al = Hhh %) (u (2) - ()) which is equal to (2). Similarly, updating ht to calculate the 1th bit A LE E [2, T as h = 1h' - and then evaluating of as af = H(1 g') we obtain af = (1), V E (1.)).
6'
A.3 Efficacy of by Propagation
A.3.1 Spatial Complexity
During the SNN inference, the bayer-b-layye propagation scheme incurs significantly lower spatial complexity compared to its alternative step-by-step propagation. This because in step-by-step inference, the computa- tions are localized at single time step for all the layers, and to process a subsequent time step, all the data, including the outputs and hidden states of all layers at the previous time step, can be discarded. Thus, the
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Operation
Bit Precision
Energy (pJ)
32
3.1
Mult.
8
0.2
32
0,1
Add.
8
0.03
32
0.13
Left. Shift
8
0.024
32
0.08
Comparator
0.03
Table 6: Comparison of the energy consumed by the different operations in our proposed IF neuron model, and multiplication required in ANNs, on an ASIC (45 nm CMOS technology). The energy values for multiply operations are sourced from Horowitz (2014); Gholami al. (2021), while those for the shift operation are from You et al. (2020): Sekikawa and Yashima (2023). The comparator energy numbers are obtained from our in-house circuit simulations, all within 45mm CMOS technology.
A=
A=2e-9
A=1e-8
2=5e-8
95.5
75
65
Test Accuracy
95.0
70
55
80
94.5
65
45
(e) VGG16
(b) ResNet?0 on CIFAR10
(c) VGG1é
(d) ResNelP0
94.0
on CIFAR10
70
60
on CIFAR100
35
on CIFAR100
93.5
55
25
4
2
No. of time steps
No. time steps
No. of time steps
No. time steps
Figure 6: Comparison of the test accuracy of our conversion method for different values of the regularization cocfficient X.
spatial inference complexity of the step-by-step propagation is O(N I.), which is not proportional to T. In contrast, for layer-by-layer propagation, the computations are localized in a single layer, and to process a subsequent layer, all the data of the previous layers can be discarded. Thus, the spatial inference complexity of the layer-by-layer propagation scheme ONN Since << T. for deep and ultra low-latency SNNs, the byyer-b-llayer propagation scheme has lower spatial complexity compared to the step-by-step propagation.
A.3.2 Latency Complexity
When operating with step-by-step propagation scheme, let us assume that the 1 layer requires (xte to process the input s² (t) and yield the output st().. Then, the latency between the input X and the output (T) is Datep = T EE= (!).
With layer-by-layer propagation, let us assume that the delay in processing the layer i.e., outputting the spike outputs for all the time steps (1)(1) VI E [1,T]) from the instant the first spike input 51 is received, is (layer()). (1). Then, the total latency between the input X and the output (TT) is Dtayer = E/-1 (I).
Although each SNN layer is stateful, the computation across the different time steps can be fused into a large CUDA kernel in GPUs when operating with the Even on neuromorphic chips such as Loihi (Davies et al., 2018), there is parallel processing capability. All these imply that Maaee (I) < T tetep (1) for any layer I. This further implies that Dtayer = E'= tlayer (l) < EL T (1) D Dstep-
propagation scheme (Fang et al. 2023
For concrete example, consider network with L 34 layers (e.g,, ResNet-34) running at T = 4 time steps. Suppose cach layer's per-step compute time (1) ~ unit (normalized). Then step-by-step latency would be Dstep 4 31 136 units. Now assume that with fused time steps, each layer can be processed in
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A=
A=2e-9
A=le-8
A=5e-8
0.8
Spiking activity
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2
4
2
3
(a)
time steps
No. of time steps
steps
(d)
No. time steps
Figure 7: Comparison of the spiking activity of the SNNs obtained via conversion method for different values of the regularization coefficient. À for (a) VGG16 on CIFARIO, (b) ResNet20 on CTFARIO, (c) VGG16 on CIFAR100, and (d) ResNet20 on CIFAR100.
34 (aayo (1) x 2 units (accounting for parallel overhead less than 4). Then Duaye 2 - 2 2 68 units, half the latency of seep-by-step. Even in more conservative scenario where (laye (1) is, say, 3 units (not fully parallel), Dlayer would be 102 units, still ~333% faster than Dstep. These numbers illustrate that throughput (in terms of latency per inference) is not sacrificed by layer-by-layer propagation; it can in fact improve it. Our method inherently exploits this by requiring only a small T, making fusion feasible. We also did not. observe any degradation in empirical throughput on GPU during our experiments the network with T ran efficiently, all operations were implemented as matrix multiplies and thresholding that GPUs handle in batch.
In conclusion, the layer-by-layer propagation scheme is generally superior both in terms of spatial and latency complexity compared to the step- by-step propagation, and hence, our method that requires layer-by-laye by-leer propagation to operate successfully, does not incur any additional overhead.
A.4 Latency& & Energy Efficiency Details
Our proposed IF neuron model incurs the same addition, threshold comparison, and potential reset operations as that of traditional IF model. It simply postpones the comparison and reset operations until after the input current is accumulated over all the T time steps. Thus, IF model has similar latency and energy complexity compared to the traditional IF model. Morcover, our proposed conversion framework roquires that the putput of each spiking convolutional layer is left-shifted by (I-1) at the 1 time step. However, as shown in Fig. 5 the number of left-shift operations in any network architecture is negligible compared to the total number of addition operations (even with the high sparsity provided by SNNs) incurred in the convolution operation. As a left-shift operation consumes similar energy as an addition operation for both 8-bit and 32-bit fixed point representation as shown in Table G the energy overhead of our proposed method is negligible compared to existing SNNs with identical spiking activity. Moreover, the energy overhead due the addition, comparison and reset operation in our (this holds true for traditional IF models as well) IF model is also negligible compared to the spiking convolution operations as shown in Fig.
Our SNNs yield high sparsity, thanks to our fine-grained L regularizer, and ultra-low latency, thanks to our conversion framework. While the high sparsity reduces the compute energy compared to existing SNNs, the reduction compared to ANNs significantly high. This is because ANNs incur multiplication operations the convolutional layer which is 6.6-31x more expensive compared to the addition operation as shown Table G Thanks to the high sparsity (71-79%) due to the E regularizer, and the addition-only operations in our SNNs, we can obtain a 1..11 reduction in the compute energy compared to an iso-architecture SNN, assuming sparsity of 50% due to the ANN ReLU layers.
The memory footprint of the SNNs during inference is primarily dominated by the read and write accesses of the post-synaptic potential at each time step (Datta et al. 2022 Yin et al. 2022). This is because these memory accesses are not influenced by the SNN sparsity since cach post-synaptic potential the sum of keen weight-modulaat spikes. For a typical convolutional layer, k 3, Gin 128, and so is almost impossible that all the kecu, spike values are zero for the membrane potential to be kept unchanged at
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Model
Time Steps
(T)
Latency per Sample (ms)
Accuracy (%)
4
8.2
95.82
Our Converted SNN
32
59.86
95.81
4
7.8
93.96
CCFF-Converted SNN
32
59.63
95.54
Table 7: Comparison of latency and accuracy between our converted SNN and QCFS-converted SNN different time steps, with the latency being profiled on NVIDIA Jetson Orin NX.
Method
Network
QQP (%)
SST-2 (%)
QNLI (%)
QCSF
ANN
84.04
81.44
80.92
QCFS
SNN
79.12
77.89
75.30
Ours
SNN
82.04
80.29
78.33
Table 8: Comparison of our proposed ANN-to-SNN conversion framework with QCFS for the BERT BASE network on a few representative GLUE tasks.
particular time stere] Since our proposed conversion framework significantly reduces the number of time steps compared previous SNN training methods, it also reduces the number membranc potential accesses proportionally. Hence, we reduce the memory footprint of the SNN during inference. However, it is hard to accurately quantify the memory savings since that depends on the system architecture and underlying hardware implementation.
Due to the exponentially lower number of time steps, our SNNs also incur lower latency compared to existing ANNs trained using ANN-to-SNN conversion. as shown in Table 7 our converted SNN model at T=4 achieves ~7.3 lower latency compared the QCFS-converted SNN at T-32, while maintaining similar accuracy. Note that our neuron model maintains the same latency as the traditional IF model, as demonstrated by the comparable latency between SNN and the QCFS-based SNN at identical time steps. These latency results are based on profiling inference per sample on the NVIDIA Jetson Orin GPU.
A.5 Performance- Energy Trade-off with Bit-level regularizer
We can reduce the spiking activity of SNNs using our fine-grained & regularizer. In particular, by increasing the value of the regularization co-efficient À from 0 to 5e-8, the spiking activity can be reduced by 2.5-4.1x for different architectures on CIFAR datasets shown in Fig. 7 However, this comes at the cost of test accuracy, particularly for a very low number of time steps, T'<=3, as shown in Fig. 6 By carefully tuning the value of A, we can obtain SNN models with different sparsity- scenarios with diverse resource budgets. Using 2=1e-8 8 for the CIFAR datasets, and >=5e-10 for ImageNet, yields good tradc-off for different time steps. As shown Fig. 2 yiclds accuracies that are similar to A= Note that A=0 implies training of the source ANN without our fine-grained regularizer for Tzelog2Q for CIFAR datasets. In particular, with ResNet18 for CIFAR10, x=1e-8 yields SNN test accuracies within 0.2% of that of À = 0, while reducing the spiking activity by ~2.4 (0.53 to 0.22), which also reduces the compute energy by a similar factor. With ResNet34 for ImageNet, À = 5e - 10, leads to 0.4% reduction in test accuracy, while reducing the compute energy by 2x. Moreover, as shown Fig. 7 the spiking activit of our SNNs trained with non-zero values of X do not increase significantly with the number of time steps as that with X=0, which also demonstrates the improved compute efficiency resulting from our regularizer.
trade-ofls that can be deployed in
A.6 Evaluation of Proposed Framework for Transformer Models
We also evaluate our ANN-to-SNN conversion framework on the BERT BASE model as shown in Table S We replace the GeLU activation function in the BERT model with the QCFS activation function to train the ANN, modified the SNN IF neuron model as proposed in our method. Note that, unlike CNNs, BERT
Note that the number of weight read and write accesses can reduced with the spike sparsity, and thus typically do dominate the memory footprint of the SNN
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Architecture
Method
ANN
T=2
T=4
T=6
7=8
T=16
7==2
SNM
74.13%
71.80%
SNNC-AP
77.89%
73.55%
OPI
76.31%
60.49%
70.72%
74.82%
VGG16
BOS 76.28%
76.03%
76.26%
76.62%
76.92%
QCFS
76.28%
63.79%
69.62%
72.50%
73.96%
76.24%
77.01%
Ours 76.71%
72.39%
76.71%
76.74%
76.70%
76.78%
76.82%
OPI
70.43%
23.09%
52.34%
67. 67.18%
BOS" 69.97%
64.21% %
65.18%
68.77%
70.12%
ResNet22
QCFS
69.94%
19.96%
14%1
49.20 19.20%
55.37%
37%
69 69.82%
Ours
70.30%
63.80%
70.30%
70.33%
70.45%
70.49%
70.52%
Table 9: Comparison of our proposed method to existing ANN-to-SNN Conversion approaches CIFAR100 dataset. Q-16 for all architectures, and x=le - 8. *BOS incurs 4 additional time steps to initialize the membrane potential, so the total number of time steps is T>4.
models do not have any batch normalization layer that succeeds the linear layer (unlike convolutional layer CNNs); and hence, we could not eliminate the unevenness error by shifting any bias term. However, our modified neuron model outperforms the existing QCFS based conversion method by ~2.8% on average for a range of tasks in the General Language Understanding Evaluation (GLUE) benchmark as shown below, We use T = 16 for reasonable trade-off between accuracy and latency.
Dataset
Approach
Architecture
Accuracy
Time steps
DSR
BPTT
ResNet18
73.35
4
Diet-SNN
Hybrid
VGG16
69.67
TEBN
BPTT
ResNet18
78.71
IM-Loss
BPTT
VGG16
70.18
5
RMP-Loss
BPTT
ResNet19
78.28
4
SurrModu
BPTT
ResNet18
79.49
Our Work
ANN-SNN
ResNet18
79.89
4
Table 10: Comparison of our proposed method with SOTA BPTT and hybrid training approaches on CIFAR100 datasct.
A.7 Comparison with SOTA for CIFAR100
We compare our proposed framework with the SOTA ANN-1o-SNN conversion approaches on CIFAR100 in Table g Similar to CIFAR10 and ImageNet, for ultra-low number of time steps, especially T<4, the test accuracy of our SNN models surpasses existing conversion methods. Moreover, our SNNs can also outperform SOTA-converted SNNs that incur even higher number of time steps. For example, the most recent conversion method, BOSQ reported test accuracy of 76.03% T=6 (with 4 time steps added top of T = 2 in Table g for the extra 4 time steps required for potential initialization); method can surpass that accuracy (76.71%) at T-4.
Additionally, as shown in Table 10 our ultra-low-latency accuracies are also higher compared to direct SNN training techniques, including BPTT and hybrid training step at iso-time-step. For example, our method can surpass the test accuracies obtained by the latest BPTT-based SNN training methods (Guo et al. 2023a Deng al. 2023) by 0.4-1.6%, while significantly reducing the training complexity.
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Dataset
Architecture
Neuromorphic
QANN
Bit-Serial
VGG16
1x
4.98x
3.57x
CIFARIO
ResNet18
1x
5.70x
4.54x
VGG16
1x
4.52x
3.12x
ImageNet
ResNet34
1x
5.12x
3.70x
Table 11: Comparison of normalized estimated energy of our SNNs on neuromorphic hardware compared quantized ANNs (QANN) and bit-serial ANNs.
A.8 Comparison with Bit-Serial Quantized ANN
Bit-serial quantization is a popular implementation technique for neural network acceleration. It is often desirable for low precision hardware, including in-memory computing chips based on one-bit memory cells such as static random access memory (SRAM) and low-bit cells, such as resistive random access memory (RRAM). Similar to the SNN, it also requires a state variable that stores the intermediate bit-level computations, however, unlike the SNN that compares the membranc state with threshold at cach time step, performs the non-linear activation function and produces the multi-bit output directly. However, to the best of our knowledge, there is no large-scale bit-scrial accelerator chip currently available. Moreover, unlike neuromorphic chips, bit-serial accelerators do not leverage the large activation sparsity demonstrated in our work, and hence, incur significantly higher compute energy compared to neuromorphic chips. Since our SNNs trained with our bit-level regularizer provides a spansity of 68-78% for different architectures and datasets, they incur 3.1-4.5x lower energy when run on sparsity-aware neuromorphic chips, compared to bit-serial accelerators, as shown Table 11 Note that the values Table [] are derived by summing compule and memory energy based on established SNN energy models, as shown in the Section A.8 of Datta et al. (202)). Specifically, estimate the number of floating point operations (FLOPs) and memory accesses in the SNN and apply per-ooeper-ooootiionn values from previous ASIC studies.
It can be argued that our approach without our bit-level regularizer leads to results similar to bit-serial computations. However, naively applying bit-serial computing to SNNs with the left-shift approach proposed in this work, would lead non-trivial accuracy degradations. This is because unlike quantized networks, SNNs can only output binary spikes based on the comparison of the membrane potential against. the threshold. Our proposed conversion optimization (bias shift of the BN layers and modification of the IF model) mitigates this accuracy gap, and ensures the SNN computation is identical to the activation-quaantzed ANN computation. This leads to zero conversion error from the quantized ANNs, and our SNNs achieve ident ical accuracy with the SOTA quantized ANNs.
A.9 Memory Bandwidth Reduction with Proposed Method
Maintaining membrane potentials across time steps significantly impacts memory bandwidth usage in Spiking Neural Networks (SNNs). Each neuron's membrane potential must be read and updated at every time step; thus, increasing the number of time steps proportionally increases memory accesses, leading to higher bandwidth consumption. Conversely, reducing the number of time steps directly lowers the frequency of these accesses, thereby decreasing memory bandwidth demands.
For instance, in an SNN with T 16 time steps, cach neuron's 's membrane potential is accessed and updatod 16 times per inference. Reducing T' to 4 decreases the number of accesses by factor of 4, effectively reducing memory bandwidth consumption by 75%. In VGG16 network designed for ImageNet classification with approsimately 6.3 million neurons, this reduction translates to fourfold decrease in memory accesses, assuming computation per time step remains constant.
The overall impact on memory energy depends on the hardware dataflow, particularly how weights and activations are stored and accessed. In weight-stationary dataflow, where weights remain on-chip and are reused across multiple inputs, memory energy is often dominated by membrane potential accesses, as they must be dynamically computed at each time step. Consequently, reducing time steps can yield memory
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Epochs
Architecture
Type QCFS pre-training
Accuracy
300
VGG16
95.82%
30
VGG16
ReLU pre-training + QCFS fine-tuning
95.47%
300
ResNet20
QCFS pre-training
93.60%
30
ResNet20
ReLU pre-training + QCFSS fine-tuning
93.51%
Table 12: Comparison of ANN training between QCFS pre-training and ReLU pre-training followed by QCFS fine-tuning for ANN-to-SNN conversion.
energy savings approaching the same factor as the time step reduction, making it a critical optimization for energy-efficient SNN implementations.
A.10 Dependence Training ANNs using QCFS Activation
While our ANN-to-SNN conversion framework based on the QCFS activation function, it cannot be dircctly applied to ANNs trained using the ReLU function. However, our experimental results demonstrate that we need to fine-tune the ANNs with the QCFS function for only a small number of epochs when they are pre-trained with the ReLU function. In particular, shown in Table [2 for both VGG16 and ResNet20, only need 30 epochs of fine-tuning with the QCFS function for ANNs pre-trained with the ReLU function to achieve the same accuracy as training with the QCFS function for 300 epochs (as done in our original experiments).
A.11 Extension to Object Detection
Model and Dataset: We chose a single-stage object detector architecture similar to RetinaNet with a ResNet-34 backbone (pre-trained on ImageNet) and detection heads for bounding box regression and classification. We evaluated on the MS COCO dataset, a standard benchmark for detection. The baseline ANN (ResNet34 backbone with feature pyramid and prediction heads) achieves an mAP (mean average precision) of around 30.1% on COCO validation this expected for a ResNet34-basse detector (which is less powerful than ResNet50-basee models that reach higher mAP)
Conversion Process: We applied our conversion framework to this detector. The backbone's convolutional layers all have batch normalization, which we adjusted (shifting the BN biases as per our method), The detection heads (for classification of proposals and for regression of box coordinates) do not contain BN layers in our implementation, but they do use ReLU activations. We converted those ReLUs to spiking neurons as well, using our proposed integrate-and-fire model. For the classification logits (which go into a sigmoid or softmax for object presence), we let the spiking neurons accumulate over T time steps and take the firing rate the analog value. For the bounding box regression outputs, which are real-valued, treated the regression approach is to represent them in fixed-point spike rate well. In our conversion, outputs as separate channels that were small in magnitude and could be encoded with a few time steps of spiking (essentially a minor extension to handle regression we discretized the output range into few spike counts). Importantly, no retraining of the detector was done after conversion; we directly used the ANN's trained weights.
straightforward
We set = time steps for the SNN detector, consistent with our low-latency focus. This means each layer of the backbone and heads processes 4 spike cycles. Our modifications (like postponed resets) ensure that this T' = 4 captures the necessary precision (in effect, equivalent to 4-bit activation quantization for the fcatures). The converted SNN detector achieves an mAP of 28.7% on COCO, which is remarkably close to the ANN's 30.1%. The drop of about 1.4 percentage points is modest and can be attributed mostly to the slight quantization of activation values due to having only time steps. Comparing to recent works on spiking neural network detectors, such as a spiking RetinaNet achieved via direct training (Kim et al., AAAI 2020) that got mAP in the 20-30% range on COCO, our converted model's mAP 29% on par with the state-of-the-art. SNN detectors reported in literature (Zhang et al. 2023), despite not doing any additional
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SNN-specific training tricks. This is strong indication that conversion is viable path to bring SNNs into complex tasks like detection with minimal performance loss.
Model
Time Steps (T)
mAP on COCO Validation (%)
Baseline ANN (ResNet-34 RetinaNet)
30.1
QCFS ANN
28.7
Converted SNN
2
27.5
Converted SNN
28. 7
Converted SNN
29.3
Table 13:
Object detection performance comparison between the baseline ANN and converted SNNs with varying time steps (T).
Model
Time Steps (T)
Noise Accuracy (%)
Adv. Attack Accuracy (%)
QCFS ANN
88.7
85.2
Converted SNN
2
86.0
83.5
Converted SNN
4
86.5
83.0
Converted SNN
6
87.0
82.8
Table 14: Robustness performance comparison between the bascline ANN and converted SNNs with varying time steps (T).
A.12 Robustness Analysis of the Proposed SNN
We conducted a brief experiment gauge how our SNN handles random noise perturbations on input images, in comparison to the source ANN. On the CIFAR-10 test set, we introduced additive Gaussian noise the images (zero-mean, with a standard deviation equal to 10% of the image pixel range). As shown in Table 14 the baseline ANN's accuracy dropped from 95.8% to 88.7% under this noise. Our converted SNN (ResNet20, T 4) saw a very similar drop, from 93.6% to about 86.5%. The gap between ANN and SNN performance remained roughly the same ( 2 3% difference) under noise. This suggests that our conversion does not degrade the inherent robustness of the model: the SNN reacts to noisy inputs just as the ANN does. some cases, we observed the SNN was slightly more tolerant to mild noise likely because the spike thresholding can filter out very small fluctuations (a tiny input change might not cause an extra spike if it doesn't it push the membrane over threshold). However, if the noise is significant, both ANN and SNN will be affected proportionally.
The enhanced robustness may be attributed of our SNNs may be attributed to the temporal integration, spike discretization, and inherited characteristics from the source ANN. Temporal robustness arises as neurons integrate information over T time steps, allowing transient noise to be averaged out, similar to a voting mechanism. While the effect is limited for small T, it still provides resilience against fleeting input perturbations. Binary robustness stems from the ol-ne--ooe nature of spikes-minot minor input changes below the threshold do not affect firing, unlike continuous activations in ANNs, which respond proportionally to small variations. Adversarial robustness largely depends on the source ANN; if the ANN is adversarially trained, this robusiness is preserved in the converted SNN (Ozdenizei and Legenstein, 2024). While conversion alone does not inherently improve resistance to adversarial attacks, SNN discontinuities and temporal dynamics can complicate gradient- based adversarial manipulations, making some attacks loss effective.
A.13 Comparison with Binary Networks
Binary and ternary networks often experience a significant drop in accuracy, especially in tasks like ImageNet classification, due their roducod representation capacity. Recent studies (Zhang et al. 2022 Bothge et al. 2021 Zhijun Tu and Wang 2022 Xu et al.) 2021 Shi al.) (022)) have attempted optimizations to mitigate this accuracy loss, but they still fall short of full-precision networks, remonstrrted in Table 15 In contrast,
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Network
Type
Dataset
Accuracy
Normalized CE
Sakr al.
2018
BNN CIFARIO
89.6
1x
Diftenderrer and Kailkhura
2021
BNN
CIFAR10
91.9
1x
Ours
SNN
CIFARIO
95.82
1.15x
Zhang et al.
2022
BNN
ImageNet
73.4
1x
[Bthhe et al.
2021
BNN
ImageNet
71.0
1x
Ours
SNN ImageNet
75.12
1.18x
Table 15: Comparison of our proposed ANN-to-SNN conversion method with T=4 against binary quantization approaches
Architecture
T
Version
Accuracy (%)
PyTorch 94.21%
2
Lava-DL 94.15%
VGG16
PyTorch 95.82%
4
Lava-DL 95.61%
PyTorch 96.12%
2
Lava-DL 95.77%
ResNet18
PyTorch 96.68%
4
Lava-DL
96.02%
Table 16: Comparison of the accuracies of our SNN models in PyTorch and Lava-DL.
our SNNSs with T-4, even achieving spike sparsity of 80%, consistently exceed the accuracies achieved by state-of-the-art binary networks.
Binary Neural Networks (BNNs) replace costly Kultiply-Accumulatt (MAC) operations with more affordable pop-count operations, leveraging binary weights and activations. While this reduces energy consumption compared to SNNs with multiple time steps, which involve additional operations per lime step, the substantial spike sparsity enabled by our l regularizer further mitigates this energy usage. Additionally, many state-of- the-art BNNs aim to enhance their expressive capacity through network modifications (c.g., ReactNet with PReLU), which significantly increase Floating Point Operations (FLOPs) by more than 2x compared to our simpler VGG and ResNets architectures. This trade-off between the combined factors of multiple time steps and additional operations versus pop-count operations and high spike sparsity in SNNs positions our SNN with T=4 to consume only 15-18% 18% more compute energy than statc-of-the-at BNNs, as depicted in Table 15 The compute energy is estimated from the energy model developed in Datta et al. 2024
A.14 Deployment of Proposed SNN on Loihi
Table 16 below compares the accuracies of our SNN in PyTorch and Lava-DI. We observed an average accuracy drop of approximately 0.3% on CIFAR-10 across both VGG and ResNet architectures when using the Lava-DL implementation compared to the PyTorch version. This discrepancy likely due to the quantization of the weights and synaptic inputs inherent the Lava-DI, framework, which introduces slight computational differences. These results are included in the revised manuscript to provide detailed analysis of the impact of deploying the SNN model Loihi via Lava-DL.
A.15 Pseudo code of proposed conversion framework
In this section, we summarize our proposed ANN-to-SNN conversion framework in Algorithm 1, which includes training the source ANNs using the QCFS activation function and then converting to SNNs.
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Algorithm :
Proposed ANN-1G-SNN conversion algorithm
Inpuls: ANN model JANN (a; W, (4, 7) with initial weight W, BN layer running mean μ4, running variance 0, learnable scale 7, and learnable variance 8; Dataset D; Quantization step L; Initial dynamic thresholds X; Learning rate e; Number of SNN time steps T
2: Output: SNN model fSNN (a; p, 3) & output s st(t) VtE(1, T"] where L = F layers
3: #Source ANN training
4: for e to epochs do
for length of datasel do Sample minibatch (a y) from D
for 1 = to fANN layers do
a' = =CFS(T+ W'd'---' of + 3*)
8:
at' = bhh starting from MSB, of the 7th term in af end for loss CrossEntropy(dd v) + A AE Et= az
10:
11:
12:
for 1 = to fANN do
13:
W -- W
Bloss
SWT,
n¹ f- 11¹
Bloss Oy
n¹ t- 0¹
Bloss do!
14:
7 + 7 e
Bloss Oh
B f 31
D3
X' &- X
Bloss ax
15: end for
16: end for
17: end for
18: #ANN-LO-SNN conversion
19: for 1= to FANN layers do
20:
JSNN W f w SNNN of
ANN X. JSNN 4t' t- PANN 3 PSNN
ANN & +(1 2) ANN JAWN
LSNNN o' JF--ANN
21:
JSNN at pANN of f SNNN 3t
JANN
22: end for
23: #erfform SNN inference on input a
24: a' = QQCF [SNNN.) .y' (= w
) + fSNN_BB .81)
JSNN
25: (t) thi of a starting from MSB
26: for 1 = 2 to fSNN layers do
27: for t= = 1 to T do
21(1)= (SNN o
(2N- w ( +SSNN_B .gt)
28:
JONN
29:
end for
(1(1) Σi=1"'
30:
31:
for
= 1 to T do s' s'(t) (u
59NN,0
32:
33:
(t 1) ut) s((t)
SNN
34: end for
35: end for
28


Reviews:
[{"review_id": "HLMw2wvphs", "reviewer": "Reviewer_vBf2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n1. The writing is great, especially the theoretical analysis of conversion errors (Section 4).\n\n2. The experiment results are remarkable, achieving 75.12% on ImageNet with only four time steps.\n\n3. It is a novel method to conduct the bias shift within batch normalization layers.\n\nWeaknesses\n1. The results for ResNet-34 trained with the proposed method in Table 2 seem unusual. Specifically, why is the performance for T=6 significantly lower than for the other conditions? Theoretically, a larger number of time steps should always lead to better performance. Could you provide an analysis of this phenomenon? Additionally, it would be helpful to include standard deviations alongside the results for clarity.\n\n2. Is the proposed conversion method applicable to training spiking Transformers, such as Spikformer [1] and Spike-driven Transformers [2][3]? Please discuss the potential for extending your method to larger SNN architectures.\n\n[1] Zhou Z, Zhu Y, He C, et al. Spikformer: When Spiking Neural Network Meets Transformer[C]//The Eleventh International Conference on Learning Representations, 2023.\n\n[2] Yao M, Hu J, Zhou Z, et al. Spike-driven transformer[C]. Advances in neural information processing systems, 2023, 36.\n\n[3] Yao M, Hu J K, Hu T, et al. Spike-driven Transformer V2: Meta Spiking Neural Network Architecture Inspiring the Design of Next-generation Neuromorphic Chips[C]//The Twelfth International Conference on Learning Representations, 2024.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a novel approach to converting Artificial Neural Networks (ANNs) to Spiking Neural Networks (SNNs), addressing the high time steps and spiking activity typically associated with conversion techniques. The framework modifies the standard integrate-and-fire (IF) neuron model in SNNs without altering the computational complexity. The proposed method produces lossless SNNs with low latency and energy consumption due to fewer time steps and high spike sparsity, while maintaining high test accuracy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See Weaknesses.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ziGfzeE0tf", "reviewer": "Reviewer_7Cxz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "While the article is generally well written, the strong and broad claims made by the authors (\"extreme energy efficiency\", \"ultra-low latency\") are insufficiently supported and the article requires significant revision before publication.\n\nStrengths: \nGenerally well written. Authors provide extensive accuracy comparisons and ablation studies.\n\nWeaknesses:\nNo convincing case is made for the strong claims in the abstract and conclusion. For example, it is apparently never stated how the authors arrive at the quantity \"compute energy\" they use to support their claims, or how the bottom-line energy for inference on a single sample would reduce based on the proposed conversion. Reductions in latency are only shown in a narrow sense: the number of time steps in the model can be reduced while maintaining accuracy. It could be argued that the original ANN has then the lowest possible latency (one time step). Authors never discuss how these \"ultra-low latency\" SNNs could translate into lower wall-clock latency, and what computational substrates would be required. It is unclear how the authors arrive at some numbers (e.g., table 6 and 10). Formatting issues like superfluous parentheses around citations (see requested changes) suggest carelessness in preparing the manuscript.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors show that quantized artificial neural networks have an equivalent description as a discrete-time multi-layer Perceptron, which is characterized as a \"spiking neural network\". It is suggested that the resulting discrete-time system offers advantages in terms of latency and energy efficiency. A regularization term to encourage sparsity by reducing the number of active bits is proposed and shown to be effective. Authors provide extensive ablation studies and comparisons with existing work, demonstrating superior accuracy.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Requested changes to content:\n\n* The manuscript claims \"extreme energy efficiency\" (6.5). However, it seems that the actual claim is that the proposed changes to the IF model to make it equivalent to the ANN do not incur a significant computation overhead relative to the original IF model. Authors should make it clear these claims regarding energy efficiency are made relative to other SNN models, and not on some absolute scale.\n\n* Authors propose an exact mapping of quantized ANNs to SNNs. Under what circumstances does this provide a \"free lunch\" in terms of energy consumption? In other words, how can energy usage be reduced while the computation is fundamentally equivalent?\n\n* Claiming \"low latency\" without further qualifiers suggests to the reader that the models can be used for faster inference based on wall-clock time. Authors should discuss what computational substrates for edge computing devices are required to actually achieve low(er) latency using lower numbers of timesteps in the models. \n\n* Table 6: authors cite four sources as well as their own research (\"in-house simulations\"). Which numbers are obtained from which sources? The caption suggests that the numbers are based on a single 45nm ASIC.\n\n* Table 10: How do you arrive at these numbers?\n\n* Figure 5: What exactly is \"spiking activity\" and how is it determined for QCFS?\n\n* Authors claim reductions in \"compute energy\", but never define this quantity. Authors need to provide a proper definition and cite sources using the same quantity.\n\n* Authors should define the spaces that parameters and variables are part of (i.e., vectors, scalars etc)\n\n* The proposed SNN bears no resemblance to biological models, but is formally equivalent to a discrete-time multi-layer Perceptron. However, what distinguishes SNNs from ANNs is precisely that spikes are discrete events in continuous time. Authors should reflect whether it is appropriate to call such a model an SNN, even if this use of the term is consistent with previous literature, and state explicitly what properties of their model set it apart from quantized ANNs or ANNs with binary activation functions.\n\nStrongly suggested changes to form:\n\nAuthors should reconsider using the term \"theorem\" for mathematical statements with proofs that consist of rather trivial algebraic manipulations and substitutions. It seems more appropriate to use \"proposition\".\n\np. 7: Fig. 3, use different colors instead of different markers. The different markers are hard to differentiate visually.\n\np.8: Fig. 4, make the dashed lines much thicker. Right now it is hard to see them and differentiate from the grid lines.\n\np. 3, remove parentheses in citations: \"In particular, (Guo et al., 2023a) and (Guo et al., 2022a)\"\n\nSuggested changes to form:\np.1: \"However, since the binary spikes in the SNN layers need to be approximated with full-precision\nANN activations for accurate conversion, the number of SNN inference time steps required is high.\" -> \"However, since the  full-precision\nANN activations need to be approximated by binary spikes in the SNN layers, the number of SNN inference time steps required is high.\"\n\np. 3: \"Note that clip(x, 0, µ) = 0, if x < 0; x, if 0 ≤ x ≤ µ; µ, if x ≥ µ.\" Write this using {cases} environment for clarity\n\np. 6: \"In contrast, (Liu et al., 2022)\" -> lower case \"in\"\n\np. 17: When repeating equations, repeat the equations using the same number instead of a new number.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DTaOAJrmIU", "reviewer": "Reviewer_67NH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n- The theoretical foundation is rigorous, with detailed mathematical proofs showing how the proposed modifications eliminate conversion errors. The authors formally prove that their approach achieves exact matching between ANN and SNN activations under certain conditions.\n- The experimental validation is comprehensive, covering multiple network architectures (VGG16, ResNet18/20/34, MobileNetV2) and datasets (CIFAR-10/100, ImageNet). The results consistently demonstrate superior accuracy-efficiency trade-offs compared to both ANN-to-SNN conversion and direct SNN training methods.\n- The proposed L1 regularization technique for encouraging spike sparsity is novel and effective. The authors show 2.5-4.1x reduction in spike activity while maintaining competitive accuracy, which directly translates to energy savings on neuromorphic hardware.\n\n### Weaknesses\n- The layer-by-layer propagation requirement of the proposed method could potentially limit parallel processing capabilities. While the authors argue this has minimal impact on modern hardware, a more detailed analysis of the throughput implications would be valuable.\n- The comparison with quantized ANNs could be more extensive. While the authors show energy advantages over 4-bit quantized networks, comparisons with more recent extreme quantization techniques (binary/ternary networks) are missing.\n- The memory overhead analysis is somewhat limited. While spike sparsity reduces compute, the impact of maintaining membrane potentials across time steps on memory bandwidth isn't thoroughly quantified.\n- The robustness of the method to different initialization schemes and training hyperparameters isn't extensively explored. Additional experiments showing sensitivity to these factors would strengthen the paper.\n- The current approach requires retraining or fine-tuning ANNs with the QCFS activation function. While the authors show this can be done efficiently, direct conversion of pre-trained ReLU networks would be more practical for real-world adoption.\n- The analysis of conversion errors focuses mainly on image classification tasks. Evaluation on other domains like object detection or semantic segmentation would better demonstrate the method's generalizability.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a ANN-to-SNN conversion framework that achieves error-free conversion with significantly reduced time steps compared to existing approaches. The key contributions include a modified IF neuron model that postpones spike firing until after input accumulation, a batch normalization bias shift technique, and a fine-grained L1 regularizer to encourage spike sparsity. The authors demonstrate state-of-the-art accuracy on image classification tasks while requiring only 2-4 time steps, representing a major efficiency improvement over prior work requiring hundreds of time steps.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See my weakness part.", "broader_impact_concerns": "N/A.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a novel approach to converting artificial neural networks (ANNs) into spiking neural networks (SNNs). It demonstrates that quantized artificial neural networks can be described equivalently as discrete-time multilayer perceptrons, which can, in turn, be characterized as spiking neural networks. The resulting discrete-time system offers advantages in terms of latency and energy efficiency. The mathematical proofs related to ANN-to-SNN conversion errors presented in the paper are intriguing, and the method shows high classification performance with only a few time steps, underscoring its utility. Although the initial version of the paper lacked sufficient support for the claims of “extremely energy efficiency” and “ultra-low latency,” these concerns have been addressed through the authors’ responses and revisions. Ultimately, the reviewers decided to accept the paper, and the AE agrees with the reviewers. There are some minor comments regarding the formatting of the references, which should be corrected in the final version.
Decision: Accept (poster)