Advancing Training Efficiency of Deep Spiking Neural Networks through Rate-based Backpropagation
Authors: Chengting Yu
Lei Liu
Gaoang Wang
Erping Li
Aili Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: Spiking Neural Networks
Training Method
Training Efficiency Optimization
Primary area: neuroscience_and_cognitive_science

Abstract: Recent insights have revealed that rate-coding is a primary form of information representation captured by surrogate-gradient-based Backpropagation Through Time (BPTT) in training deep Spiking Neural Networks (SNNs). Motivated by these findings, we propose rate-based backpropagation, a training strategy specifically designed to exploit rate-based representations to reduce the complexity of BPTT. Our method minimizes reliance on detailed temporal derivatives by focusing on averaged dynamics, streamlining the computational graph to reduce memory and computational demands of SNNs training. We substantiate the rationality of the gradient approximation between BPTT and the proposed method through both theoretical analysis and empirical observations. Comprehensive experiments on CIFAR-10, CIFAR-100, ImageNet, and CIFAR10-DVS validate that our method achieves comparable performance to BPTT counterparts, and surpasses state-of-the-art efficient training techniques. By leveraging the inherent benefits of rate-coding, this work sets the stage for more scalable and efficient SNNs training within resource-constrained environments.

Advancing Training Efficiency of Deep Spiking Neural Networks through Rate-based Backpropagation
Chengting Yu!.2, Lei Liu", Gaoang Wang?, Erping Li Aili Wang 1.2
1 College of Information Science and Electronic Engineering, Zhejiang University
2 *ZJU-UIUC Institute, Zhejiang University chengting.o. Srnggiing.1@iing 21@intl...... faj.nnl.ju...... www..du..u...................... edu. en, ailivangeintl www. edu.
Abstract
Recent insights have revealed that rate-coding is primary form of information representation captured by surrogate-gradde
Backpropagation Through Time (BPTT) in training deep Spiking Neural Networks (SNNs). Motivated by these findings, we propose rate-based backpropagation, a training strategy specif- ically designed to exploit rate-based representations to reduce the complexity of BPTT. Our method minimizes reliance on detailed temporal derivatives by fo- cusing averaged dynamics, streamlining the computational graph to reduce memory and computational demands SNNs training. We substantiate the ra- tionality of the gradient approximation between BPTT and the proposed method through both theoretical analysis and empirical observations. Comprehensive experiments CIFAR-10, 10, CIFAR-100 100, ImageNet, and CIFARIO-DVS validate that our method achieves comparable performance to BPTT counterparts, and surpasses state-of-ther the-art efficient training techniques. By leveraging the inherent benefits of rate-coding, this work sets the stage for more scalable and efficient SNNs training within resource-constrainee environments. Our code is available https://github.com http://ittubb.oo////aaa//tttttte -attratteebaae-d-ee
baspeeebbaabbbaaapppaaattiii backpropagation
Introduction
Spiking Neural Networks (SNNs) conceptualized as biologically inspired neural systems, incor- porating spiking neurons that closely mimic biological neural dynamics [46] (56). Unlike Artificial Neural Networks (ANNs) based continuous data representations, SNNs adopt spike-coding strate- gies to facilitate data transmission through discrete binary spike trains
(52] The intrinsic binary
mechanism eliminates the need for the extensive multiply-acuumulate operations typically required for synaptic connectivity [56], thereby enhancing energy efficiency and inference speed when deployed
on neuromorphic hardware systems 10 54]
The mainstream training methods for SNNs primarily utilize Backpropagation Through Time (BPTT) with surrogate gradients to overcome di-differentialle spike events, allowing SNNs to achieve comparable results with ANNs counterparts [51] 62 74]. However, the direct training method necessitates the storage of all temporal activations for backward propagation across the network's depth and duration, leading to high training costs in terms both computational time and memory 47 13). To alleviate memory burdens, online training techniques have been developed that partially decouple the time dependencies of backward computations in BPTT However, online methods still require iterative computations based on the time dimension, increasing training time complexity as the number of timesteps grows.
demands [43 86
35
77
76
76 48 89
"Corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Observed across most biological sensory systems, rate coding is a phenomenon where information encoded through the rate of neuronal spikes, regardless of precise spike timing [52 64 23]
Recent explorations into spike representation have demonstrated the significant role of rate coding in enhancing the robustness of SNNs, further confirming its dominant position as the encoding representation in networks [38 60 [8] A significant observation has shown that BPTT-trained SNNs on static benchmark exhibit spike representation primarily following the rate-coding manner by highlighting strong similarities in representation between SNNs and their ANN counterparts [44]. similar conclusion resonated with findings in fields of adversarial attacks, where recent methods significantly benefit from based representations to enhance attack effectiveness 66 29 50].
Motivated by rate coding's s status as the most effective and predominant form of representation in SNNs, we posit that targeted training based on rate-based information could offer a high cost- effectiveness ratio. We propose to decouple BPTT based on rate-coding approximation and simplify rate-based derivative computations to a single spatial backpropagation. We further provide theoretical analysis and empirical evidence to reveal the rationality of the gradient approximation between BPTT and the proposed method. Experimental results demonstrate that the proposed method achieves performance comparable to BPTT counterparts while significantly reducing memory and computational demands. Comparison results also indicate that the proposed method outperforms state-of- thetheeart efficient training methods on benchmarks. We expect our work to facilitate more efficient and scalable training for SNNs in resource- condurrieddd environments. Our main contributions are as follows:
We propose rate-based backpropagation that leverages rate-coded information for efficient training of deep SNNs. This method simplifies the computational graph by decoupling and compressing temporal dependencies, reducing training time and memory requirements.
Alongside the proposed method, we conduct theoretical analysis and empirical validation to demonstrate its effectiveness in approximating the gradient computations performed by BPTT-based SNNs training.
We conduct experiments on CIFAR-10, CIFAR-100, CIFARIO-DVS, and ImageNet, demon- strating that our proposed method matches the comparable performance of the BPTT counterpart and achieves state-of-the-art results among efficient SNN training methods.
2 Related Work
Training Methods for Deep SNNs. Deep SNNs are trained primarily through two principal strategies:
(1) conversion methods that establish links between SNNs and ANNs through equivalent closed-form mappings, and (2) direct training from scratch utilizing Backpropagation Through Time (BPTT). Conversion methods develop closed-forn formulations for spike representations [39 67 72 88
73 47,, enabling seamless transitions of pre-trained ANNs into SNNs and facilitating comparable 57
12 42 16.. However, the precision of these mappings under ultra-low latency conditions is not consistently reliable, often necessitating extensive time steps to accumulate spikes, which may compromise performance 7 40 31 28
performance large-scale datasets 88 15 27
59
34]. Direct
training methods permit SNNs' performance with extremely low time steps by employing BPTT along with surrogate gradients to compute derivatives of discrete spiking events [5T] 62 74 22
81 87]
[3]. The strategy fosters innovation in SNN-specific modules, including optimized neurons, synapses, and network architectures, thereby enhancing performance [25 20
85
43
666
69
77 83
24
80
61.. Despite the advantages of low latency, direct training imposes substantial
memory and time burdens maintain the backward computational graph 43 86
35
77
76
To mitigate training costs associated with direct methods, light training strategies have attracted
86 55 70 Several studies have explored the concept of decoupling the forward and backward passes in SNNs. which generally assumes that neuronal dynamics follow deterministic processes and aims to establish closed-form fixed-point equivalences between spike
considerable attention 49 35
representations and corresponding rate-based activations 722 73 777 47 68]. Drawing on online training techniques from recurrent neural networks, several studies have adapted the principles of Real-time Recurrent Learning (RTRL)
[ to streamline the online training process for SNNs, aiming to decrease memory demands while preserving biologically plausible online properties of the The online methodologies have proven effective in large-scale tasks 76 89] Nevertheless, the significant time costs associated with training methods continue
networks [84
2 82
55
76
47 89
to challenge SNNs' broader application.
2

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Forward Pass
Forward Pass
Forward Pass
Loss
Backward Pass
Loss
Lass
Loss
Loss
Bax Bockwarrd
Ouly
Online
7 #Haaee
s
$2+3
(overt)
1
ut
M4-11
W
W
W
Time T
Gradian Computation
Time
Memory Optimization
Time T Memory Time Optimization
Backward Memory; O(LT)
Backward Memory: ((L)
Backward Memory: O(L)
Backward Time: OLLT)
Backward Time: (LT)
Backward Time: (L.)
(a) Standard BPTT Training Figure I: Illustration of the forward and backward procedures of different training methods.
(b) Online Training
(c) Rate-based Backpropagation
Spike Coding in SNNs. SNNs transmit information through spike trains [52] with encoding
mechanisms classified into temporal and rate coding. Temporal coding is defined on firing times,
employed by several direct trainings [49 75 (8) and ANN-to-SNN conversions [26 65 is noted for its low energy consumption due to sparse spiking. However, temporal coding schemes often require specialized neuron configurations and are generally effective only on simpler datasets 26
65
88
Conversely, rate coding is widely adopted across both conversion 12] 15
16
27
36
57
59 78 and
direct training approaches [73 (7747 47. consistently achieving superior performance and facilitating
low-latency operations [77] 477. Morcover, rate coding has demonstrated significant potential in [8]. with attack methods specifically designed to exploit rate- based representations showing promise in surpassing benchmarks for SNNs defense against attacks (6. 30 (5). By employing representation similarity analysis to compare BrTTetrained SNNs with their ANN counterparts, Li et al.
enhancing the robustness of SNNs against adversarial attacks [38 60
44 has indicated that rate coding serves as the primary mode of information representation [44). Inspired by previous findings, we consider that rate-coded information represents the most effective and predominant form of signal expression in SNNs, and the targeted training based rate-based spike representations may offer high cost-effectii cesstfffffctieeesss ratio. Therefore, we propose to decouple BPTT towards rate- baseb backpropagation with the purpose of enhancing the efficiency of SNNs training.
3 Preliminaries
3.1 Spiking Neural Networks
Inspired by the brain's s ability to transmit information through discrete spikes, the Leaky Integrate- and-Fire (LIF) model serves as the basic building block of SNNs due its simplicity. For practical implementation of SNNs based on connected spiking neurons, the dynamics of the LIF model are typically rendered in discrete iterative format:
u = Auu - Vin - w's =i Vin)
(I)
where vi and sf represent the membrane potential and output spike of neurons in layer / at time I, respectively. W' denotes the linear synaptic connections between layers / - 1 and l, and À acts as the decay term for the membrane potential. The Heaviside step function, H-)), determines spike generation, ensuring st in binary forms. Noting that H-) ) is not differentiable, SNNs' direct training employs surrogate gradients to achieve error propagation by creating various pseudo-derivatives following the basic idea of Straight-Througe Estimator (STE) (3]
ST 74 19],
3.2
Training SNNs with BPTT
The network outputs at each timestep t are given by o, W where W denotes the classifier's weights. Classification is based on the average of these outputs across all timesteps, computed as Upeed 1 E Or, The loss function L is defined over averaged outputs and is typically formulated as -= ET
where y represents the true labels and l could be the cross-entropy BPTT unfolds the iterations described in Eq. (1) and propagates gradients back along the computational graphs across both temporal and spatial dimensions, as illustrated in Fig. 1.. The gradients of the membrane potential incorporate elements
function, noted in various studies 88
48
19 69

---

Forward
Rate-based Representation
based Backpropagation
dc
T
Örl
$ 12-1
22
S [++1
0s
0s;
0s
0.c
0.C
00
24
2
U
0u
duG
0u
1
c w't
dcl
W
Wi
W
ac
WH
Pate-Coding App Poxoommatio
52
1-1
c-1 c+11
8pl-1
backward Figure 2: The implementation of rate- backpropagation across layers. A rate-coding approxıma- tion utilized for the forward procedure to connect average inputs with rate outputs, enabling fast rate-based error backpropagation throughout the training process.
from both (spatial) spike generation and (temporal) potential accumulation, expressed as:
ac ac ask Oul
OC dulyi
(Dul.1 (ōu!
Oul+1 as! Osh ou:
5sk ou
Ou!
(2)
Oc 0s
Oc ast
@dutt du,
@ul+ Ost Os! ou!
II
ast
dul
TSt
1=T-1
Subsequently, the weight update for layer is determined among timesteps T, i.c. VwiL=
E
Oc Bu,
E'=
8C
s 1-1T and the gradient is further propagated to previous layers through
Tu OW!
ou!
the linear part by
Oc
DC W
0s;
I-
ou!
Rate-based Backpropagation for SNNs Training
4.1 Derivation of Rate-based Backpropagation
Incorporating rate-based representation. Under the rate coding assumption, essential information effectively encapsulated within the spike frequency averages. We start by defining the rate- based representation approximation for the forward procedure in SNNs, shown in Figure The average firing rate at each layer I, denoted as r', is calculated as the expected value of the spike outputs s over the temporal dimension rd E[
Eigr s4
Considering the forward propagation through linear operators with weights W that compute the inputs as ! w's! instead of transmitting distinct spikes over multiple timesteps, transform the average rates into average inputs c' in the approximate representation:
c' = EIT E s W'Bls ''
Supposing input representations are well captured within c², we approximate the exact inputs with the average inputs for all timesteps, I ss ct, and follow the neuronal dynamics in Eq. ① to derive the output rates r' = [[s(]] With the rate-coding approximation in place, we can derive the gradients with respect the weights in the linear part based on the error propagated through the average inputs:
Oc dc
dc 1-1T dc
V Li rate dc Ow
Handling temporal dependency during backward. For back back-propagating the error, the linear parts Be! operate smoothly as लр Wt The next step is to define the correlation between the averages of inputs and output spike rates, Ber within the neurons of layer Since there is no deterministic relationship between r' and c², we first look into the influence of separated inputs following the exact
dr

---

gradients in Eq.
0s 01
as
Ou
06
II
i=7-1
(0ul4s iuu
+
Suiys 08) 0s; ou!
if 2 if = if <
De
(4)
Oui
Ou:
By accumulating the intricate dynamics over time, can derive the gradients of the overall spikes with respect to the inputs at time l:
Os aI;
dst Or
Ost Oul
@ui+1
an Os! ds; du!
II
(5)
Ou!
i=r-1
Here, with rate-coding approximating I c², we follow the idea of StraightTThrough Estimator @r: and define the backward gradients as oet = Id, with Id representing the identity matrix. Then, we can derive the surrogate gradients of neural dynamics through the mean estimator:
3
Or' Dc rale
Σ
O(E [s:] ar. 01 dc
1 ΣE T
ds E [
(6)
Or.
With the compressed gradients of neuron parts, the error backpropagation of the rate-based represen- tation then determined, dependent only on the spatial domain:
ac ac
Oc
II i=L-1
Oct+1
Ōr dc )mmem
))
Oc
II (w E (w"TB(Z)) i=L-1
rate
a
Ori
d
where we define the objective L -((E(o)) - ee(ct, y). Note that the rate-based representa- tion, while instrumental in constructing the backward computational graph for learning, does not necessitate actual implementation during the forward pass.
4.2 Rate-based Gradient Computation for Memory and Time Efficiency
As previously discussed, rate-based backpropagation can executed on spatial-dimensio com- putation by decoupling BPTT. We now show how rate-based backpropagation can be efficiently implemented within the overall learning framework. As depicted Figure [p. online schemes apply eligibility traces e, locally within neurons to store historical information, effectively block- ing backward access to past gradients. The gradient computation optimized by compressing all past temporal dependencies into ei. Similarly, utilize iterative variables ( and {ee hsz as the accumulated post- and pre-synaptic dependencies, synchronously recorded during the neu- ral dynamics computations. The iteration of {cl his dynamically records the firing rates, where el = ((t 1)e(-1 + s)), and it is straightforward to derive r' ety. Considering the surrogate @r' gradients of neural dynamics, de to estimate future-dependent terms outlined in Eq. 5 we first construct equivalent eligibility trace forms, (p' with iterative expressions starting at p 1:
( Ouk Ou, 0sk-1 Du
(Jui
dulti Oul Os! du!
p = 11 P
1+ Σ
II TCt 1=111
(8)
Dst_ ou
Os! du!
with the equivalence that:
0s5
ds!
Oul+1 + Oul+z Os!
ds! ou! ))
Σ x Σ
+
II
Orl!
Ou
au!
T>t
imr-1
Üs!
+ auk.. as! @ultt1 Oul
ds! ou. 1)
Os! Orey pi:
Σ
+
II
On!
rsti=t1
By iteratively accumulating (
del auf pt). obtain gf 0o! Bu!
p\] E E[xx]] Now,
we have collapsed the required computation graph through the iterative calculation to complexity OLL). The rate-based propagation is then conducted in go, relying only the intermediate variables e/ g/., and W! within one-time spatial-dimension backpropagation.

---

4.3 Connecting Error Backward of Rate-based Backpropagation to BPTT
Having derived the fundamental form of rate-based backpropagation through the rate-encoding approximation, we now explore potential divergences with BPTT during error propagation. Although rate-based hackpropagation is derived from the approximated forward pass, it still provides valid gradients for the original network parameters.
The primary divergence between rate-back and BPTT in backward computation primarily arises from the assumptions regarding the approximation of rate-based representation through mean estimators, as outlined in (3) and (.) The rate-coding motivations establish equivalence with BPTT by assuming temporal components are independent, which is formalized in Theorem
Theorem 1. Given 515²)
ac that refers to gradients computed following the chain rule of BPTT in ös;
Eq. 22, and n². = =
ae or.
(where E
[m:]] = E
[xx]] in Eq. 6 7 i
ki = E
0
E
[+]]]
holds for Vl. we have E [o!'0
DC 8r
Furthermore, given 8.
oc
if
6
rate
ar!
[lss/""),00 Vl, we then obtain ( + (/V.L)). Here, E [ax]] + E, xe refers
the mean value of tensor over temporal dimension T.
To confirm our hypotheses, we carried out empirical experiments, the results of which are detailed in the experimental section. Our empirical findings support the core assumptions outlined in Theorem
demonstrating the relative independence between (ee)) and kcj (Figure Bd,), as well as between 8! ((r)) and sy (Figure 3p). For minor discrepancies that may arise, we introduced Theorem 2, which tolerates small deviations and confirms that approximation errors in rate-based backpropagation can be effectively bounded, ensuring the robustness of training under practical conditions.
Theorem 2. For gradients 8(8')
ac de;
and k2 En
given the approximation error bound JĀ
[) Vl. Denote the stacked tensor and st == [s,...., str). Assuming the backward procedure follows non-expansivity s.t. W4+1 as
E (()))
E (()))
[+e]] || S c11 + ||
<>
S.I.
E
F'
[T
OF is I-lipschitz. continuous without loss of generality and the biases are bounded uniformly by B, i.e.
ar OF
1 or 00r-11
- |fr Vax i. Define 8'm
02 as the
31
error propagated through Eq.
7), and 8/4))
oc as the error propagated through BPTT, with or;
8 = E,0!"" We have the gradient difference bounded by 0 E66! 1 O(kee)
Theorem elucidates the stability of rate-based backpropagation relative to BPTT, showing that the proposed method can provide a bound on the overall objective solution. The bounded error could further be interpreted as form of randomness suitable for stochastic optimization. The similarity measurement of the descent directions between the two methods provides empirical evidence for the effectiveness of the proposed method (Figure 3d.. Detailed proof is provided in Appendix A.
4.4 Implementation Details
For implementations of direct training, two distinct training modes are recognized: (multi-step) activation- based and (single-step) time-based [19], differing fundamentally in handling the timesteps loop. We implement our rate-based propagation both modes: rate denotes the multi-step training mode where T loops are embedded within layers, and rates refers to the single-step training mode with T loops outside the layers. A detailed discussion of training modes is included in Appendix B.
Another aspect of our implementation concerns handling batch normalization (BN), especially given its critical role in BPTT, which adjusts mean and variance statistics during the forward pass. The application of BN varies depending on the training mode. In the multi-step mode, BN benefits from access to information across all timesteps and can normalize based on statistics aggregated over temporal dimensions. We employed tdBN [87] in rate since it has been widely adopted in direct training on various benchmarks. In contrast, the single-step mode limits BN to current timestep inputs, necessitating normalization across spatial dimensions only. In line with online schemes, SLTT 48 demonstrates the feasibility of implementing spatial BN iteratively across timesteps, an approach adopt for rates. Further details on the BN implementation are provided in Appendix B.

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Meesurement of
cos Similarryy
(b) Mesesmeneee A
Magnitude 11--5
mo LONG
AL: 1.18mmm-3
TG 1:8312
1--164m-125585
Berdiions
di
8k 24 04
(0) Meoourementt o2
Cosine Similerto.
(0) Maanrrmmenn of
cosige Similarity
LBBCC
16 L48313
tte Serition 4
8k
24
8k
Ieeraione 2k
Figure 3: Empirical measurements conducted on the training procedure of BPTT. The experiments
are carried out on the CIFAR- 100 dataset using ResNet-18, Each subplot is labeled according to the
naming convention "A (test#)-T [timesteps#)- [Larget)-L[ B (block# IN (LI## [CC /CCI
Experiments
In this section. we conduct experiments on CIFAR-I0 (37. CIFAR-100 [3,, ImageNet [I]]. and evaluate the proposed training method. We implement SNNs training on
CIFARIO-DVS 4T
the Pytorch [53 and SpikingJelly [19] frameworks. We set Van 1, À = 0.2, and employ the
sigmoid- based surrogate function 19 for LIF neurons. Detailed setups are provided in Appendix C.
5.1 Empirical Validation
Empirical experiments are conducted to support the preconditions of theorems discussed in Sec- tion 4.3. These preconditions assert the independence of paired variables across the temporal 66 x2!
dimension: E
(A1) and E 6, s. E 5 (') Eis!- (A2). To explore these relationships, we conducted experiments training ResNet-18 on CIFAR-100 using BPTT. Cosine similarity measures were employed compare the empirical expectation products,
E 6!0² E [<]]]
(B (6(²) m!]
0 E [<x]]) as shown in Figure 3., where values approaching I indicate high
EE
degree of alignment, suggesting that the variables' directions are similar. Additionally, the correlation cov(.s..
coefficient, P was measured to further assess the independence of these variables p v var(re, )aan v(r( o(("))
where COV(K,, 0. = E 0 x2 - E 8
E [<]]] It is clear that P equals the cosine distance
between the variables after centering by their means, p cos( - ) n²t E [x(])) Re- sults, shown in Figure Bp. reveal that the correlation coefficients are constrained within a very small range. typically around the magnitude of 10 supporting the hypothesis of their rela- tive independence. We also conducted cosine similarity measurements to validate the assumption ((("))(-11 =E
"] as shown in Figure Additionally, we implement both BPTT and the proposed method simultaneously within the same training iteration, allowing direct observa- w L) (A3) was visualized in Figure 3d. which revealed that the convergence directions for rate-based backpropagation and BPTT are closely aligned. Remarkably, all tests consistently demonstrate that configurations with T=6 better adhere to the theoretical assumptions than T=4, suggesting that the proposed method can more closely mimic BPTT computations as the timestep increases. This observation also highlights the intrinsic link between method and rate-coding, suggesting that larger temporal window may facilitate more stable manifestations of rate-coding.
tion of the gradient descent directions. The relation WV = + (Vw
5.2 Comparison with the State-of-the-Art
We present comparison results in Table I] In single-step mode, rates offers fair comparisons with online schemes, while rafe multi-step mode competes fairly with other methods employing one-step backpropagation. Unlike online methods such as OTTT[76), SLTT(48), and OS[89]. which necessitate spatial backpropagation at every timestep, our proposed method conducts this process only once at the final timestep. Although methods of DSR 44 and SSF[68] delay decoupled backpropagation until the final timestep, allowing for parallel processing across all timesteps to enhance computational speed, they still require each timestep's 's backpropagation to be managed independently within the backward computation graph. In contrast, our method fully compresses the temporal dimension, achieving one-step time-independent spatial backpropagation. As shown in Table [] our method yields comparable performance with BPTT counterparts on benchmarks, showcasing promising capabilities compared to other efficient training methods. While our theoretical
7

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Table I: Performance on CIFAR-1O, CIFAR-100, ImageNet, and CIFARIO-DVS. Results are averaged three of experiments, except for single crop evaluations ImageNet. Models marked with (*) employ scaled weight standardization, adapting to normalizer-free architectures.
Training
Method
Model
Timesteps
Top-I Ace (%)
QCFS
7
ANN2SNN
ResNet-18
8
94.82
DSR
47
one-step
PreAct 18
20
95.10:10.15
SSF 68
one-step
Prs-ct-ResNet-88
20
94,90
BFTT
ResNet-18
4
95.64
ratef (ours)
one one-step
ResNet-18
95.61
CIFARIO
OTTT 76
online
VGG-11"
93.52:::0.06
SLTT
online
ResNet-18
94.442:0221
VGG-11
94.35
0S R9
online
ResNet-19
4
95.20
ResNet-18
4
95.53
BPTTs
RPTT
VGG-11
95.61
ResNet-18
42:42220111. (195.556)
rates (ours)
one-step
VGG-11
95.57=10.08(95.66
DSR
one-step
PreAct-ResNet-le
20
78. 78.50:+:.12
SSF 68
one-step
PreAct-ResNet-18
20
75.48
ВРTТ
BPTT
ResNet-18
4
77.93
rates (ours)
one
step
ResNet-18
4
78. 0. 12788.3))
OTTT 776
online
VGG-11'
6
71.05-t:0004
CIFAR100
SLTT 48
online
ResNet-18
6
74.38:10.30
VGG-11
76.48
OS
89
online
ResNet-19
77.86
BPTTs
RPIT
ResNet-18 18
4
77.72
VGG-11
4
77.82
ResNet-18
4
77. 77.73:10.28(77.99
rates (ours)
one-step
VGG-11
77.88--0.35(88.1)
OTTT 76
online
PreAct-ResNet-34
65.15
SLTT 48
online
PreAct-ResNet-33
66.19
NiW-ResNet-B33
64.14
OS 89
online
ImugeNNet
PreAct-ResNet334
67.54
SEW ResNet
20
HPTT
#EW-ReNNet-34
4
67.04
SEW-ResNet-34
65.66
rates (ours)
one-step
PreAct-ResMet-34
69.58
SEW ResNet 34
65.84
rate (ours)
one-step
PreAct-ResMet-34
70.01
DSR 47
VGG-11
20
77.27-+0.24
68
one-step
VGG-11
20
78.0
CIFARIO-DVS
76
VGG-11"
76.63++-.354
online
SLTT 48
VGG-11
10
77.17-40.23
BPTTs
VGG-11
10
76.73
BPTT
ВРTТ
VGG-11
76.86
rates (ours)
VGG-11
76.48:10.23(76.71
rate (ours)
one-step
VGG-11
10
76. 76.96:10.13//7.133
analysis and motivation primarily adhere to rate-coding approximations, the performance on static datasets aligns with expectations. The results on the dynamic dataset CIFARIO-DV'S also achieve comparable levels, implying significant presence of rate- based representation within CIFARIO-DVE More results regarding the performance comparisons between the proposed method and BPTT across various architectures and settings have been detailed in Appendix D.
5.3 Impact of Time Expansion
we assess the impact of extending timesteps on both accuracy and training efficiency. Figure validates that our method capably manages increased timesteps, thereby confirming the scalability of the proposed method for larger T values. Figure 4 displays the computational and memory expenses incurred during the backward phase, which, as anticipated, do not escalate with increasing T.
8

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Memory (GS)
Top-1 ACC (8)
95
CIFAR-10 CIFAR-10
4
2
2
4
8
10
12
Time (ms/batch)
79
0.3
rate
BPTT
78
77
CIFAR-10 100
0.1
6
2
10
12
Timesteps T
Timesteps
(a) Comparison of Classification Performance
(b) Comparison of Training Costs
Figure 4: Results of BPTT and rate across various timesteps.
Table 2: Performance w/o and w/ temporal shuffle for models trained by rate
Dataset
Model
Timesteps
Accuracy
Shuffied
2
94.77
94.6320.04
ResNet-18
95.51
95.50+0.04
95.97
95.95-0009
CIFAR-10
2
95.13
95.10=0.055
VGG-11
95.37
95. 95.37=00.03
95.77
95.79+00.05
76.277
15.5
ResNet-18
78.32
77.72.10.15
79.10
79. 79.10110.14
CIFAR-100
2
77.46
77.21=0.12
VGG-11
77.88
77.78=0.16
77.97
78.02+00.99
SEW-ResNet-St
65.84
65.11+0.11
ImageNet
PreAct ResNet-34
69.78.10.10
CIFARIO-DVS
VGG-IT
76.50
14.69
5.4 Analysis of Rate Statistics
Our method, derived from the principles of rate-based representation, necessitates examining the impact rate coding on model behavior. Following an insightful approach from [6), we assess the robustness of our models by shuffling the temporal order of spike sequences while maintaining their rate consistency. This experiment, designed to disrupt temporal information without changing the firing rate, was applied to models trained using rate-based backpropagation. During inference on the test dataset, we introduced perturbations by randomly shuffling the temporal dimensions of input tensors across all neurons, reported in Table 2 Notably, models mostly resisted these changes to some degree, which suggests that they follow the basic rules of rate coding, where the reordering of timesteps does not significantly impact overall accuracy. Furthermore, rates across each layer over time, presented in Figure As layers increase, the average spike rates per layer closely aligned with the temporal mean, validating the idea of rate-coding approximation. Those two experiments support the notion that rate-based backpropagation proficiently captures rate-based representations during training.
tracked the average firing
0.4
t=1
t=1
Firing Rates
MM
0.4
0.3
t-3
tw3
0.3
0.2
90.2
tw5
0.2
mean
0.1
0.1
10
30
16
Layer Index
Layer Index
(a) Rate Statistics of SEW-Resnen Resnet-34 on Imagenet
(b) Rate Statistics of Resnet-18 on CIFAR-100 Figure 5: Firing rates statistics for models trained by rate
Conclusion
In this work, we propose rate-based backpropagation, utilizing rate-coding approximation to stream- line the gradient computational graph, significantly reducing both memory usage and training time. Through theoretical analyses and empirical validation, we show method's feasibility in approxi- mating the optimization direction of BPTT. Experimental results across benchmarks reveal that our

---

method achieves comparable performance with BPTT and surpasses other state-of of-the--rttet efficient training methods. We expect our work pave the way for more scalable and resource-efficien training of SNNs.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant No. 62304203), the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ22F010011), and the ZJU-UIUC Center for Heterogeneously Integrated TrainnInspired Computing.
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15

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A Proof of Theorems
Theorem 1. Given 610²)
ac that refers to gradients computed following the chain rule of BPTT in
Eq. 22) and rel = En
0st
(where E3
[nc]]
E
[x]]] in Eq. 6 7)
if
0
x/] = E
di
E
[mc]]
holds for vl, have I.
0C
Dr
race
Furthermore, given 6.
a7."
8C
1/
5,
8
1-1
0
- for Vl, we then obtain (Vwi (VWiL Here, E
[x/]
+ E,
De refers
the mean value of tensor over temporal dimension T.
Proof. Given 6/0²)
0e' 9C and k²t Ex
we establish the mean gradients through neural aτ
Deg
dynamics based on the chain rule:
ac 11
OL ast,
T Σ
OL 0s
= E[5 mx]).
E
= E
(10)
Üst or
Üs 0r,
Considering the output layer 1 = L. the objective for BPTT can be expressed as L. = L y) [f = E(
[WLBB(s)) y) = E(WL,d L yy) Under the rate-based objective
e((c 4) ((( 3). it is clear that E[o;" [ (art of 1) ) Applying the precon-
dition El xx]] E(o
E 10 11 we obtain:
OL 0rt-1 --2
0c
E[rc
IW(1-))
Orl-1 dcL 0rL-2
OTL-1
rite
rate
EE 6
[E
E|8(("11) (1) RC
1 (-I]((1-1))
0C
(11)
EE
W
=E 01!
W(L-1)
01.-
1-1
OC
El
2)1.
=E
[as4-2
Continuing this induction process, we can derive that B|8 (0r) rate for all layers l. Further, ac ar! aw!
given (12)) 0c the gradient for the weight matrix under BPTT: VwiL= E.
at!-
Σ. 0: (7)) s/_1 The gradients passing through the linear parts maintain the equivalence:
dc Oc
OC 0r'+1
OC Inl+I
W4+1
Onl
Oc
rate
(12)
OL
WL+ = =E[5(e")
W w 1= 1E(
=FE
Örl+1
With the precondition that E[5!" (')
s E(8!" EEss 1-1
holds for Vl, we obtain:
( aC dc
OL
(V w: L) Dr rate
I-1
Oc aw?
dc!
rate
rite
(13)
8 =R () "] TE(o(1 (1')
V
WIL
T
Theorem 2. For gradients 5. ac
and k²} = E-
a1! given the approximation error bound
0s
|| Vl. Denote the stacked tensor I' = [r..., and s = is..... s/r. Assuming the backward procedure follows non-expansivity s.t.
[<]]] S <c( |[ [o
<> S....
E
8
kc - 8
E
0r'+1 wi+1
a is I-lipschitz continuous without loss of generality and the biases are bounded
de
ar
uniformly by B, Lee. a ar't1
4 0r'
< x & for Vae, &c. Define 8 = ( (ee) rate as the
16

---

error propagated through Eq. 7. and "5
St; as the error propagated through BPTT, with
tn = -Bi8 )
We have the grudient difference bounded by 0
" = ok)
Proof. Given that the enror backpropagation bounded by B for all I, can derive
ar'+1 follows 1-L.isshiitz condition with biases Of
[ non-expansivity. Then, by induction, we obtain the gradient bound between the intermediate layers and the final layer:
||o
E
(1 ar'+1
||ا B by
(B16 = (L l)B + B
Since ar W R is also 1 1-L-ischitz continuous without loss of generality, given the S7. approximated approximated error > 0 s.t.
E
( wl+1" E[rc]] - E66!! w 141 x²]
||o B[rc]] - EiS n4]
(14)
$ c((1 +
E(0 m|] = +
have
5"
(1)
51+ W 1+1 E [[[c]]
Bio!
W
W
W
E[Ni]]
Elo!
|n E E[ra))
+
W E[saa]
(15)
87+1
1888
+ te(1 + |ila! wi+1 mill
81+1 miee
El5
+ c(1 + (L- B)B + 1
By induction, we obtain
(L 5' Aac (ETEEE((11 s (
B (L. (1 o (11-1))2) (16)
2
B Implementation Details
B.1 Pseudocode of the Rate-hased Kackpropagation
The pseudocode for rate-based backpropagation, illustrating the implementations for both rate and rates, provided in Algorithm
B.2 About Training Modes in Rate-based Backpropagation
In direct training, two distinct implementation modes are recognized, octivation-based and time-based 19), differing fundamentally in their handling of the simulation timestep T. The activation-based, also known as multi-step mode, processes the T loop separately within each layer, transmitting inter-layer tensors within dimensions [T, B, S), where B and S refer batch and spatial dimensions, respectively. The configuration enables the multi-step mode to enhance computational efficiency by reformatting the tensor dimensions as [T' SJ to optimize parallelism in linear parts. However, the coupled processing with temporal calculations embedded within the layers increases memory retention on GPUs, potentially obscuring the benefits of memory cost optimization in both online training and our proposed methods. In contrast, the time-based mode externalizes the T loop, facilitating single-step forward computations at each timestep. This single-step mode aligns well with the dynamic modeling of temporal dimensions and facilitates memory optimization strategies more effectively. However, restriction on parallel computation in linear components compared to multi-step mode necessitates increased forward time on GPUs, albeit with enhanced support for memory optimization. Our proposed method has been adapted to operate effectively within both frameworks to ensure comprehensiveness, as shown in Algorithm 1.
17

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Algorithm 1: Single Training Iteration of the Rate-based Backpropagation
Input: Timesteps T'; Network depth L; Trainable parameters (W") his: Training Mini-batch {(a) y)}: Training Mode rates or rate M-
Output: Updated parameters { [Wh
Initialize input spikes si = xς for all f [1.
if rate M then
for to I. do
Compute input currents through linear operators I W' s' for all E [1,T];
Initialize p = 0, g = 0, eb = O. for t to do Compute output spikes sf from I following neural dynamics in Eq.
I
Compute the eligibility trace p -1+ pi-1
Ou du: 80 Fuc
in Eq. (8)
0e
Accumulate e' = i ((t 1)e| s();
Dai Accumulate of } ((1 11)111)91111 ou. po).
10
11
end
Save e',, 96 and w for backwards, and free intermediate variables. end
12
13
14 else
Initialize p g 0, e for all / [].
16
for t = to I do
17
for / to do
Compute input currents through linear operators I W's
18
Initialize pb = 0, gó = 0, e O; Compute output spikes ot from I following neural dynamics Eq.
19
20
1
21
Compute the eligibility trace pt + p\-n
ou!_,
ou!
+ Ou!
0s.-1 0u.
80
in Eq. 8
Accumulate e ((1 1)e + s));
23
Accumulate of = ( 111-1)9h-111 ou: Pt))
0o!
Save u/, st for neuron states;
24
25
Save g e pf as eligibility traces. end
26
end
end
Compute the outputs gradient dez from the objective function. for T, do
8C Compute error backpropagated through the linear part Gee
W411
31
Bel
32
Compute error backpropagated through the neuron part
Bel
SC
8C
gr:
Compute the weight gradients V ac ae (el-t)T:
Update parameters (W') her based on the gradient-bar basee optimizer.
34
end
B.3 Implementation Batch Normalization in Rate-based Backpropagation
In the forward pass, batch normalization (BN) precedes neuron activation, scaling inputs I and introduces a bias in the average inputs = = We denote é to represent the biased average inputs as ē = E. E[))(I.) instead of Note that BN acts as a linear operation during inference, where ē = F.[ = [[)N(L.) = BN(E(T.)
BN(c). Implementing rate-based propagation requires considering how gradients pass through the BN layers and affect their intrinsic parameters during training. Initially, explore the spatial BN (48 design for the single-step mode, which
18

---

computes mean and variance statistics independently at each time step t:
= BN( =
I, Ht
+B, where μe - - x and o 1 D 54
- -m)) (17)
V0 +
Defining xi
0í, x(
81,
01.
By
Xi
((8)
af. 0g
the following expressions are obtained:
ai; d
of 0p. 0p 01
- Pr Voł+ +
x!
( = Id. (18)
Xt = Y
+
of +
do? 01,
For the backward derivation of BN in a rate- based setting based on mean estimations through time, we implement 0C
a de F
(7)]
OL
de =(x! ac Since gradient
dc
=
de
computation at each timestep is independent, the dynamic estimations of E(x: and E Eix!")
performed in the same manner of (e/her and (g|)-s--
In the multi-step mode, tdBN 87 accounts for mean and variance statistics over the entire time horizon:
f, BN(L)) =7 1 (//-- + B, where μ +c
00 0² BT EE
1 BT
ΣΣ
1 b
12)
(19)
The rate-based representation integrates the input across the time dimension, with the mean a Ee (b)) and variance a Ex(c(() :c)2 Since is the temporal mean of inputs, it is clear that p.. μ1 and a² ²². Note that å al.
E. (T, μt) (c pe) 8
Assuming 0r, = Td, we derive 8u 0μc For the forward approximation specifically tailored for ăt,
de tdBN in rate- rate-based backpropagation, we define:
de
(
غ = BN(c) -7 (V22 -e
c 1
+ B,
(20)
where 7 and ,3 refer to the same intrinsic parameters shared with BN( and o2 is defined distinctly
in forward and backward passes: a σ² in forward and Be Td in backward. The implementa-
0o
tion utilizes gradient replacement with the detach operation in PyTorch: o² detach(o² .²) +
Thus, in the forward phase, è BN(() E[)) and in the backward phase, dc E 08 aligning perfectly with the foundational principles of based backpropagation.
de
C Experimental Settings
C.1 Datasets
CIFAR-10 and CIFAR-100. The CIFAR-10 and CIFAR-100 37 datasets contain 32x32 color images across different classes, licensed under MIT. CIFAR-10 includes 60,000 images across 10 classes, with 50,000 for training and 10,000 for testing, whereas CIFAR-100 is spread over 100 classes. Both datasets have been normalized for zero mean and unit variance. Image data augmentation is applied using Augment [9] and Cutout [14] strategies, similar to the implementations in recent
The pixel values are directly fed into the input layer at each timestep as studies 42 7 24 69 13
direct encoding (55).
ImageNet. The ImageNet- dataset [T] comprises 1,281,167 training images and 50,000 validation images distributed across 1,000 classes, licensed for non-commercial use. ImageNet- K images are normalized for zero mean and unit variance. Training images undergo random resized cropping to 224x224 pixels and horizontal flipping, while validation images are resized 256x256 and then center-cropped to 224x224. The images are transformed into time sequences through direct encoding 55 following the approach used for CIFAR datasets.
CIFARIO-DYS. The CIFARIO-DVS dataset 41 a neuromorphic version of CIFAR-10 100 which includes 10,000 event-based images captured by the DVS camera with pixel dimensions expanded to
19

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Cosine Similarity
1,00
000
0.86
cas
0.00
ga-C3
A1-sim-N4
A2-sim-CS
A3 23-gd-f CS
A1-sim-N6
A2-sim-C7
A3-gd-C7
Iterations 4k
Iterations
Iterations
Figure Empirical measurements conducted on the CIFARIO IO-ARSODDV dataset.
128x128, licensed under CC BY 4.0. We split the whole dataset into 9000 training images and 1000 testing images. Data preprocessing involves integrating events into frames 21 19 and reducing the spatial resolution to 48x48 through interpolation. Additional data augmentation includes random horizontal flips and random rolls within a 5-pixel range, mirroring previous methods 76 48
C.2 Training Setup
Network Architectures. For the CIFAR-IO, CIFAR-100, and CIFARIO-DVS datasets, our method
is tested on standard network architectures, including ResNet-18, ResNet-19, and VGG-11 [63
32
87
76 19 69.. On the ImageNet dataset, we adapt two variations on ResNet architecture [2]
NEW-ResNet-33 34 [87] specially proposed for SNNs. and ResNet-34 with pre-activation residual
blocks B3 aligning with previous works 76 48 89 While OTTT (76 and SLTT [48] frameworks
utilize normalization-Fii these with their uusoo-designed batch normalization. We directly employ tdBN normalization-fra free methods in our experiments.
techniques under the ResNet-34 framework [5], Zhu et al.
89
substitute
87
instead of
Training Details. This work utilizes the widely adopted sigmoid-based surrogate gradient [19 to approximate the Hcaviside step function using h(x, a)) = Ttene and sets 4 to ensure for preventing gradient explosion. All implementations based the PyTorcc 53 and SpikingJelly [19] frameworks. The experiments on CIFAR-10, 10, CIFAR-100, and CIFAR 10-DVS datasets run on one NVIDIA GeForce RTX 3090 GPU. For ImageNet, distributed data parallel processing is utilized across eight NVIDIA GeForce RTX 4090 GPUs. We use the SGD optimizer
the maximum derivative of the surrogate function
(58) with momentum of 0.9 for all tasks, integrating cosine annealing strategy 45 for the learning rate schedule. Other hyperparameters are listed in Table 3
Table 3: Training hyperparameters.
CIFAR-10
CIFAR-100
ImageNet
CIFARIO-DVS
Epoch
300
300
100
300
Learning rate
0.1
0.1
0.2
0.1
Batch size
128
128
512
128
Weight decay
5e-4
5e-4
2e-5
5e-4
D
More Results
D.1 Empirical Validation on CIFARIO-DVS
As shown in Figure 6 we extend conduct empirical experiments on CIFAR10-DVS as avalidation in the case of dynamic datasets. The observations confirm that, even in data with degree of temporal information, the empirical validation of the assumptions remains consistent with expectations. This alignment emphasizes that the approximate relationship between rate-based backpropagation and BPTT remains substantially consistent. As result, this stability ensures that our approach continues to effectively extract rate-based representations from neuromorphic datasets with a degree of temporal dynamics, thereby maintaining robust performance across diverse data scenarios.
D.2 Extended Performance Comparisons with BPTT
We conduct additional experiments to illustrate the comparative performance of rate-based back- propagation versus BPTT, as presented in Table for CIFAR-I0 and Table CIFAR-100. These experiments span various configurations, including different network architectures Wwchieeturss-ReeNee-JJS
20

---

Table 4: Performance comparison of rate-based backpropagation and BPTT on CIFAR-10. 10.
Training
Model
Timesteps
Top-1 Acc (%)
95.02
ResNet-18
4
95.53
95.68
96.12
BPTTss
ResNet-19 19
96.38
96.57
2
95.27
VGG-11
95.61
95.63
2
94.82+00.07744889)
ResNet-18
(1.45:
95.73+0.03(95.78)
96.11110.05(9..18)
rates
ResNet- 19
96.32++0.04(963388
96.38++0.00696.455)
95.44+0.00((5.46)
VGG-11
95.57+0.08895.66
95.6420.12(95.766
94.93
ResNet-18
95.64
96.03
96.16
BPTT
ResNet-19 19
96.49
96.70
95.31
VGG-11
95.67
95.64
94.75++0055(9.82)
ResNet-18
05.61:00.02(95.64
95.90+0.07796.001)
96.23++0.1006.33)
rate
ResNet-19
96.26:10.0399..29)
96.38110.02(96440)
95.17 (
VGG-11
95.30++0.06695.377
95.23+0..0695.322)
21

---

Table 5: Performance comparison of rate-based backpropagation and BPTT on CIFAR-100.
Training
Model
Timesteps
Top-1 Acc (%)
76.24
ResNet-18
4
77.72
78.65
79.33
BPTTss
ResNet-19 19
80.12
80.77
2
77.37
VGG-11
77.82
78.13
2
75.89++0.11
ResNet-18
77.73:+028877.99
78.86+0.08(78.94)
79.71+00002(7..74)
rates
ResNet- 19
80.41 80.404
80.75++..05000779)
77.34+0.04(7.377
VGG-11
77.87
78.23:10.03(7.227)
77.09
ResNet-18
77.93
78.35
80.01
BPTT
ResNet-19 19
81.07
81.12
77.42
VGG-11
77.96
78.25
75.97:0.20(76.27)
ResNet-18
19.021)0.1(79.16)
79.89)90.03(79.90)
rate
ResNet-19
80.7140..1280.84)
80.83100.0788.994)
77.4040.00(77.466
VGG-11
77. (7.86-90.0((77.88)
71.9910.11((8.111)
22

---

Table 6: Comparison results of performance and training costs across various timesteps. All units for time measurements are in seconds per batch. Experiments were conducted on NVIDIA GeForce RTX 4090, with training settings consistent with other experiments.
Timesteps
Datasets
Network
Method
T=1
T=2
T=4
T=8
T=16
Time of Eligibility Track
0.003
0.004
0.007
0.015
0.027
Time of Backward
0.034
0.035
0.036
0.034
0.036
ratear
Time of both
0.037
0.039
0.043
0.049
0.063
Memory Allocated
1.8492
1.8488
1.8473
1.8496
1.8483
ResNet-18
Top-1 Acc [%]]
74.60
76.04
78.24
79.24
79.37
Time of Backward
0.023
0.044
0.098
0.199
0,564
BPTT
Momory Allocated
1.4272
2.4454
4.4804
8.0460
15.685
Top- Acc [S6]
74.38
76.65
78.49
78.35
Time of Eligibility Track
0.006
0.012
0.020
0.041
Time of Backward
0.083
0.083
0.082
0,083
ratess
Time of both
0.089
0.095
0.102
0.124
Memory Allocated [GB]
4.4787
4.4798
4,4788
4,4784
CIFARI00
ResNet-19 19
TTop-1 Acc [%6]
78.3
80.00
80.65
81,31
Time of Backward
0.046
0.111
0.285
0.552
BPTT
Memory Allecated [GB]
3,2556
5.6636
10.8978
20.3862
Top-1 Acc [%]
78.39
80.06
81.11
81.13
Time of Eligibility Track
0.003
0.003
0.006
0.011
0.020
Time of Backward
0.017
0.017
0.017
0.017
0,018
ratess
Time of both
0.020
0.020
0.023
0.028
0.038
Memory Allocated [GBB]
1.3624
1.3607
1.3619
1.3613
1.3601
VGG11
Top-1 Acc [%]
76.13
77.59
77.75
78.34
78.65
Time of Backward
0.010
0.021
0.054
0.135
0.384
BPTTM
Memory Allocated [GB]
0.9911
1.6784
3.7363
6.6141
12.3768
Top-1 Ace [%]
76.34
77.20
77.98
78.26
78.37
Time of Eligibility Track
0.012
0.014
0.023
Time Backward
0.074
0.074
0.074
rate ar
Time of both
0.086
0.088
0.097
34
Memory Allocated [GIB]
5.7887
5.7898
5.7883
Time Backward
0.046
0.095
0.233
BPTT M
Memory Allocated [GB]
3.9858
6.8654
12,5597
ImageNet
Time of Eligibility Truck
0.007
0.009
0.020
Time of Backward
0.072
0.071
0.072
rate
Time of both
0.079
0.080
0.092
PreAct-ResNNe-34
Memory Allocated [GB]
5.4995
5.4982
5.4942
Time of Backward
0.046
0.088
0.211
BPTT
Memory Allocated [GB]
3.7017
6.4778
11.969
ResNet- and VGG-1 timesteps (T=2, 4, 6). The results demonstrate that rate-based back- propagation maintains competitive accuracy with BPTT across different architectures and timestep settings benchmark datasets.
D.3 Comprehensive Evaluation of Training Costs
To enhance the understanding of the scalability of the proposed method, we extended our analysis to include training costs across the CIFAR-100 and ImageNet datasets, utilizing additional network architectures as detailed in Table 6 This comprehensive evaluation aimed to assess the impact of varying time steps on performance, memory, and time costs. We integrated the computation of eligibility traces during the forward process, ensuring a fair comparison by incorporating these
23

---

iterative computations into the overall cost assessment. The results reveal that the total cost of rate-based backpropagation demonstrates a clear advantage over BPTT when timesteps T which underscores the efficiency of the proposed method approach in managing computational resources while maintaining comparative performance across various datasets and network architectures.
E
Social Impacts and Limitations
There is no direct negative societal impact since this work centers enhancing the training efficiency of SNNs. SNNs inherently require less energy for inference compared to ANNs, helping reduce carbon dioxide emissions. The methods developed in this work further optimize SNNs training by improving both memory and time efficiency, potentially reducing the overall resource consumption and environmental footprint of training processes. Regarding limitations, this work primarily com- pares with BPTT baselines, and there is potential for incorporating strategies from state-of-the-ar techniques in future work. Moreover, the proposed method is tailored for tasks that utilize rate-coding, designed to efficiently capture spatial rate-based feature representations to enhance training; therefore, necessitates further adaptation effectively manage sequential tasks. Future efforts may need to delve deeper into adapting the dynamic characteristics of spikes and robustly designing training hyperparameters, ensuring compatibility with rate-basee based backpropagation and extending applicability to wider range of applications.
24

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Reviews:
[{"review_id": "bUtFAKKx5c", "reviewer": "Reviewer_TjVW", "summary": "This paper proposes a rate-based SNN training method, which can effectively reduce memory and time cost during training. They proved the efficiency of the rate-based back-propagation training and demonstrate that the rate-based training outperforms other back-propagation methods.", "questions": "The authors introduce the rate-coding approximation forward propagation in Section 4.1. Is this forward propagation method also used during inference?\n\nWhat is the performance of rate$_s$ on ImageNet dataset?", "limitations": "See in weaknesses.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The rate-based method achieves better performance and uses less computing resource compared with BPTT, which is impressive.\n\nThis paper is well-written and well-organized.", "weaknesses": "The novelty is weak. There are two previous works that share similar idea with this paper, since they all use rate-based backpropagation [1,2]. The author needs to briefly explain the differences between these papers.\n\nThe rate-based backpropagation is not suitable for sequential tasks.\n\n[1] Li, Yuhang, et al. \"Differentiable spike: Rethinking gradient-descent for training spiking neural networks.\" Advances in Neural Information Processing Systems (2021).\n[2] Bu, Tong, et al. \"Rate gradient approximation attack threats deep spiking neural networks.\" Computer Vision and Pattern Recognition (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "#### **W1: The novelty is weak. There are two previous works that share similar idea with this paper, since they all use rate-basedbackpropagation [1,2]. The author needs to briefly explain the differences between these papers.**\n\n\nThank you for your comments, which have prompted further clarification of our work's novelty and contributions.\n\nFirstly, regarding the differentiation from references [1] and [2]:\n\n1. The work in [1] focuses primarily on the calibration and modification of STE [3] (or \"surrogate-gradients\" commonly used in SNNs). In contrast, our approach significantly simplifies the chain rule application by compressing backpropagation within the spatial dimensions to enhance training efficiency.\n2. Meanwhile, [2] derives gradient relationships based on a deterministic relationship assumption between inputs and outputs in a closed-form manner, similar to the methodologies used for ANN-to-SNN conversions [4]. Conversely, our method adopts a stochastic dynamic process for gradient computation, which follows the ideas of decoupling BPTT-based direct-learning methods. Additionally, while [2] targets the adversarial attack domain where precise gradient calculations are not stringent, our proposed method as a training method requires more accurate gradient computations, demonstrated experimentally to achieve gradients akin to those of BPTT.\n\nFurthermore, other works bind ANN and SNN training together in what's known as tandem training [5]. However, these approaches cannot connect well to established BPTT, limiting their accuracy and competitiveness with BPTT benchmarks. There are also efforts considering rate-based gradient graphs [6], aimed at optimizing training costs, but these are similarly limited by closed-form derivations like [4], unable to bypass the coupling of batch normalization layers for purely spatial backpropagation.\n\nWe appreciate your suggestion for greater clarity, and we will include a detailed discussion in the appendix to better delineate how our method differs from other rate-based approaches. Thank you for prompting this improvement.\n\nTo further clarify our novelty and contribution, it is important to note that our work's originality does not claim to be the first to propose rate-based backpropagation. Rather, our significant contribution lies in being the first to effectively implement the concept of rate-based backpropagation as a training efficiency strategy, demonstrating its potential to perform comparably with BPTT. This is not intended as a perfect substitute for BPTT but as an effective complement and enhancement, forming a promising combined approach. We believe this technical solution, which optimizes the training process of deep SNNs, holds potential developmental benefits for the community.\n\n\n#### **W2: The rate-based backpropagation is not suitable for sequential tasks.**\nWe acknowledge that rate-based approaches [2,4,5,6] including our proposed method, indeed face challenges in handling sequential tasks, which is a recognized limitation of such strategies. This work does not aim to address the difficulties of applying rate-based methods to sequential tasks. Instead, our focus is primarily on tasks that rely mainly on rate-coding, and the proposed method is specifically designed to capture the rate-based feature representation of deep SNNs to enhance training efficiency. Thank you for highlighting this point; we will include it in the limitations section.\n\n\n#### __Q1: The authors introduce the rate-coding approximation forward propagation in Section 4.1. Is this forward propagation method also used during inference?__\n\nNo, the rate-coding approximation is purely used during the training phase and does not participate in the inference process. The inference strictly adheres to the conventional protocols of deep SNNs to ensure fair comparisons.\n\n#### __Q2: What is the performance of $rate_S$ on ImageNet dataset?__\n\nAdditional experiments have been conducted to evaluate the performance of $rate_S$ on the ImageNet:\n\n||Timesteps | Top-1 Acc (%)|\n|:-:|:-:|:-:|\n|SEW-ResNet-34|4|65.656         |\n|PreAct-ResNet-34|4|69.578       | \n\nThe results confirm that the $rate_S$ method is effective on ImageNet. We will integrate these results into Table 1.\n\n\n\n## Reference:\n[1] Li, Yuhang, et al. \"Differentiable spike: Rethinking gradient-descent for training spiking neural networks.\" Advances in Neural Information Processing Systems 34 (2021): 23426-23439.\n\n[2] Bu, Tong, et al. \"Rate gradient approximation attack threats deep spiking neural networks.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.\n\n[3] Bengio, Yoshua, Nicholas Léonard, and Aaron Courville. \"Estimating or propagating gradients through stochastic neurons for conditional computation.\" arXiv preprint arXiv:1308.3432 (2013).\n\n[4] Bu, Tong, et al. \"Optimal ANN-SNN Conversion for High-accuracy and Ultra-low-latency Spiking Neural Networks.\" International Conference on Learning Representations.\n\n[5] Wu, Jibin, et al. \"A tandem learning rule for effective training and rapid inference of deep spiking neural networks.\" IEEE Transactions on Neural Networks and Learning Systems 34.1 (2021): 446-460.\n\n[6] Meng, Qingyan, et al. \"Training high-performance low-latency spiking neural networks by differentiation on spike representation.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2022.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WBvXqoWskK", "reviewer": "Reviewer_YGtz", "summary": "This work falls into the category of efficient SNN training methods. This paper proposes a reduced computational graph to reduce the memory and computational demands of SNNs training. This work has the potential to train SNNs on resource-limited devices. The paper evaluates the methods on CIFAR-10, CIFAR-100, ImageNet, and other datasets.", "questions": "Why did the authors approximate the spiking rate directly with the mean over timesteps, instead of using a running mean with a decay parameter $\\lambda$, which would more closely approximate the rate in the leaky integration mode?\n\nIn Line 151, page 5, what does 𝑑 represent in $\\frac{\\partial I}{partial c} = Id$?\n\nPlease elaborate further on the differences between rateM and rateS. The authors state that 'rateM represents the multi-step training mode where T loops are embedded within layers, while rateS refers to the single-step training mode with T loops outside the layer.'\n\nRegarding robustness to $\\lambda$ In the paper, the neuronal parameter $\\lambda$ is set to 0.2. Can you provide experiments with other values of $\\lambda$, such as 0.4, 0.6, 0.8, and 1.0?\n\nI believe that the training cost in Fig. 4 should encompass not only the backward process but also the forward iteration process (which also contributes to the cost).", "limitations": "See weakness and questions.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper addresses the issue of high time and memory costs in training spiking neural networks. \n\nThis paper provides solid theoretical insights into the error bound and its relation to SNN BPTT training. \n\nThe results of this work are comparable to the performance of the BPTT counterpart.", "weaknesses": "Not a clear comparison of the differences with existing e-prop methods in terms of methodology. \n\nNo generalization results on hyperparameters (e.g., $\\lambda$) are presented in this work. I raise this question because most work on SNNs uses large values of $\\lambda$, but this work used 0.2 as $\\lambda$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "#### **W1: Not a clear comparison of the differences with existing e-prop methods in terms of methodology.**\n\nThank you for your comments. In our paper, we compare our method with online-learning akin to e-prop [1], as OTTT[2,3,4], SLTT[3], and OS[4], with results shown in Table 1. Descriptions of online methods can be found in Lines 29-32, 81-87, 161-166, and 251-255. \n\nThe core differences in implementation with these methods, as highlighted in Fig. 1b and 1c, are that: \n1. Unlike online methods that require spatial backpropagation at every timestep, our proposed method conducts this process only once at the final timestep, effectively compressing the multiple instances of spatial backpropagation into a single occurrence.\n2. Methodologically, the proposed rate-based method differs from e-prop in that we do not focus on specific temporal dynamics among all time slots but rather aim to efficiently train networks by capturing dominant rate-based features.\n\nThank you for pointing this out. We will ensure these details are more clearly articulated in the paper.\n\n#### **W2: No generalization results on hyperparameters (e.g. $\\lambda$) are presented. Most work on SNNs uses large values of $\\lambda$, but this work used 0.2.**\n\nThank you for your comment. We configured our training environment and hyperparameters based on setups from existing works on direct training-based deep SNNs [5,6,7,8,2,3,9,4,10,11]. In these works, the decay parameter $\\lambda$ is set at 0.5 in [2,4,11], 0.25 in [8], 0.2 in [7,9,10], and 0.09 in [3]. It is observed that direct training methods often employ $\\lambda$ of 0.5 or lower. We chose 0.2 to align with training settings used in [7,10].\n\n#### **Q1: Why did the authors approximate the spiking rate directly with the mean over timesteps, instead of using a runningmean with a decay parameter, which would more closely approximate rate in the leaky integration mode?**\n\nThank you for your comments.\n\nFirstly, the choice of averaging over timesteps for measuring the spiking rate aligns better with our method: \n1. Using a running mean with a decay parameter [12,3], which measures a scaled running average of the rate, typically retains temporal dimensions where weights for spikes closer in time are higher. This approach often requires constructing rate-based backpropagation separately for different time slots.\n2. In contrast, our approach involves constructing a single rate-based chain rule for spatial-only backward propagation at the final timestep. Therefore, approximating the mean rate over the entire neural dynamics is more suitable for our goals.\n\nRegarding the second question, considering the rate estimation at time $T$, our method gives $r_1=\\frac{\\sum_t s_t}{T}$, while a running mean with a decay parameter would give $r_2=\\frac{\\sum_t \\lambda^{(T-t)}s_t}{\\sum_t \\lambda^{(T-t)}}$. If the spike sequence follows frequency coding with a spike rate $r$ (assuming $s_t \\sim Bernoulli(r)$), then $\\mathbb{E}[r_1]=\\frac{\\sum_t \\mathbb{E[s_t]}}{T}=r$ and $\\mathbb{E}[r_2]=\\frac{\\sum_t \\lambda^{(T-t)}\\mathbb{E}[s_t]}{\\sum_t \\lambda^{(T-t)}}=r$. This supports that both methods provide unbiased estimates of the rate.\n\nIn summary, our choice of averaging over timesteps for rate approximation is driven by its compatibility with our single-step spatial-only backpropagation approach, simplifying the implementation and aligning with our method's objectives effectively.\n\n\n#### **Q2: In Line 151, page 5, what does $d$ represent?**\n\nWe apologize for the ambiguity; $Id$ represents the identity matrix. We will add clarification.\n\n\n#### **Q3: Please elaborate further on the differences between rateM and rateS.**\n\n$rate_M$ and $rate_S$ cater to multi-step and single-step training modes, respectively, each adapting batch normalization differently as detailed in the the Appendix B.2 and B.3. We support both modes to demonstrate method’s flexibility and to facilitate fair comparisons with related methods.\n\n\n\n#### **Q4: Can you provide experiments withother values of $\\lambda$, such as 0.4, 0.6, 0.8, and 1.0?**\nAdditional experiments on CIFAR10 with T=6 setting using ResNet-18\n\n|$\\lambda$|0.1|0.2|0.4|0.5|0.6|0.8|1.0|\n|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n|$rate_M$|95.45|95.8|95.65|95.34|95.11|93.82|92.79|\n|BPTT|95.51|95.69|95.48|95.19|95.31|94.65|94.7|\n\nThe results indicate that when $\\lambda$ is within the range of $[0.1, 0.5]$, the performance of our rate-based method aligns closely with that of BPTT. However, as it increases into $[0.8, 1.0]$, the performance of both BPTT and rate-based methods declines, with a more pronounced decrease in the rate-based method. We believe the primary reason for this phenomenon is the non-ideality in the assumption of independent distributions used in Theorem 1, $\\mathbb{E}[\\delta^{(s^l)}_t \\kappa^l_t ] = \\mathbb{E}[\\delta^{(s^l)}_t]\\mathbb{E}[\\kappa^l_t]$, which leads to larger approximation errors at higher $\\lambda$ values, as discussed in Theorem 2 where $\\epsilon$ is amplified, thus magnifying the gradient difference with BPTT. We will address this issue in the appendix, discussing the implications and acknowledging it as a limitation in our study.\n\n\n#### **Q5: I believe that the training cost in Fig. 4 should encompass not only the backward process but also the forward iteration process.**\n\nThank you for your feedback. We have supplemented our work with more detailed experiments on training costs, including the extra costs incurred during the forward iteration process as shown in the global response. Firstly, the total time cost combined with the computation of eligibility traces in forward, still shows a clear advantage over BPTT when $T\\geq2$. Additionally, it is observed that the computation of eligibility traces is less intensive than the backward process when $T\\leq 8$.\n\nWe will include the supplemented experimental results into the paper and acknowledge that the computation of eligibility traces during the forward process requires further optimization.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "COMieNpRAB", "reviewer": "Reviewer_B3hA", "summary": "This paper presents a novel rate-based backpropagation method for spiking neural network (SNNS) training, which effectively separates the time-dependent backpropagation (BPTT) process and thus reduces computational and memory costs. The method employs a rate-encoded approximation to capture the basic information and is validated by empirical experiments on various datasets, demonstrating that it is superior in terms of training efficiency and accuracy when compared to the traditional BPTT.", "questions": "1. In lines 223-234, the reference to 'when cosine similarity close to 1 is interpreted as a high degree of consistency in the direction of the variable', does it take into account the effects of data distribution and noise, which may also occur in the case of uneven data distribution. Can additional experiments or theories be added to rule out the effect of data distribution and noise on the hypothesis presented in lines 223-234?\n2. The approach proposed in the paper seems to be very similar to the one described in reference [1]. Although the general direction of the two is different, the core idea seems to be the same. Could you please explain the difference between your approach and the one outlined in reference [1]?\n3. In Section 5.3, in the experiments evaluating the effect of time step on accuracy and training, only one dataset, CIFAR-10, was used. Could the experiment be supplemented with experiments using other datasets to demonstrate the scalability of the proposed method for larger values of T?\n4. In the caption of Fig. 3, is the placeholder '#' missing from T{timesteps}?\n\nReference:\n[1] Bu, Tong, et al. \"Rate gradient approximation attack threats deep spiking neural networks.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.", "limitations": "The authors fully explain the limitations and potential social implications of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Empirical results on multiple datasets (CIFAR-10, CIFAR-100, ImageNet, CIFAR10-DVS) support the theoretical claims and ensure accuracy while reducing memory and time costs.\n2. The paper is well-written, clearly explaining the proposed method, theoretical underpinnings, and experimental validation.", "weaknesses": "1.\tIn lines 53-55, this paper mentions that the proposed method reduces training time, but there is no relevant experimental proof in the experiments section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "#### **W1: In lines 53-55, this paper mentions that the proposed method reduces training time, but there is no relevant experimental proof in the experiments section.**\nThank you for your suggestion. We have added more experiments on training costs to strengthen the experimental proof of training efficiency, as shown in the global response. These results will be included in the paper. Thank you for pointing this out.\n\n#### **Q1: In lines 223-234, the reference to 'when cosine similarity close to 1 is interpreted as a high degree of consistency in the direction of the variable', does it take into account the effects of data distribution and noise, which may also occur in the case of uneven data distribution. Can additional experiments or theories be added to rule out the effect of data distribution and noise on the hypothesis presented in lines 223-234?**\n\n\nThank you for your suggestions. \nFirst, we have supplemented empirical experiments on DVS-CIFAR10 as a validation in the case of uneven data distribution (please see Figure in the global response). The observations confirm that, even in data with a degree of temporal information, the empirical validation of the assumptions remains consistent with expectations. \nFurthermore, we would like to clarify Theorem 2 mentioned in the paper. We derived a general approximation error in Theorem 2, which provides a proof concerning the error bounds. This can demonstrate that the approximation errors caused by non-ideality of assumptions are well bounded. We believe that both the effects of uneven data distribution and noise can be generally considered as factors causing approximation errors in our assumptions, and Theorem 2 has already accounted for these scenarios.\n\n\n#### **Q2: The approach proposed in the paper seems to be very similar to the one described in reference [1]. Although the general direction of the two is different, the core idea seems to be the same. Could you please explain the difference between your approach and the one outlined in reference [1]?**\n\nHere are the main differences between our work and reference [1]:\n\n1. Our method to gradient computation differ significantly. We likely to conclude the concepts into, gradient-computing via stochastic dynamic process v.s. deterministic closed-form. [1] derives gradient relationships based on a deterministic relationship assumption between inputs and outputs in a closed-form manner, similar to the methodologies used for ANN-to-SNN conversions [2]. Conversely, our method adopts a stochastic dynamic process for gradient computation, which follows the ideas of decoupling BPTT-based direct-learning methods [3,4].\n2. The application domains of the two methods vary, influencing dinstinct objectives. The method in [1] is used primarily in adversarial attack scenarios, focusing on the input data of the entire model. In such cases, the requirement for strict adherence to the exact gradient computation process is more relaxed, and deviations are somewhat permissible. In contrast, our method is designed for training deep SNNs, necessitating more precise gradient computations for all weights. Our approach supports more strict gradients within rate-based backpropagation, and experiments have demonstrated the proposed method can achieve gradients akin to BPTT.\n\nWe will include a topic in the appendix to provide a more detailed description of how our work differs from related work. Thank you for your comment.\n\n\n#### **Q3: In Section 5.3, in the experiments evaluating the effect of time step on accuracy and training, only one dataset,CIFAR-10, was used. Could the experiment be supplemented with experiments using other datasets to demonstrate the scalability of the proposed method for larger values of T?**\n\nThank you for your suggestion. Our additional experiments evaluated the effects of timestep on accuracy, memory, and time costs across the CIFAR-100 and ImageNet datasets using various network architectures (in global PDF). These results are consistent with the discussions in our manuscript and demonstrate the scalability of the proposed method. We will include the additional results into the updated version.\n\n#### **Q4: In the caption of Fig. 3, is the placeholder '#' missing from T{timesteps}?**\n\nThank you for pointing that out; it will be corrected.\n\n\n## Reference\n[1] Bu, Tong, et al. \"Rate gradient approximation attack threats deep spiking neural networks.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.\n\n[2] Bu, Tong, et al. \"Optimal ANN-SNN Conversion for High-accuracy and Ultra-low-latency Spiking Neural Networks.\" International Conference on Learning Representations.\n\n[3] Bellec, Guillaume, et al. \"A solution to the learning dilemma for recurrent networks of spiking neurons.\" Nature communications 11.1 (2020): 3625.\n\n[4] Xiao, Mingqing, et al. \"Online training through time for spiking neural networks.\" Advances in neural information processing systems 35 (2022): 20717-20730.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "11AIRtSmav", "reviewer": "Reviewer_Fw27", "summary": "Recent research indicates that rate-coding is crucial for information representation in deep Spiking Neural Networks (SNNs) trained via Backpropagation Through Time (BPTT). Building on this insight, a new training strategy called rate-based backpropagation has been developed to leverage rate-based representations, reducing the complexity of BPTT. This approach focuses on averaged dynamics to simplify the computational graph, thereby lowering memory and computational requirements. Theoretical and empirical analyses demonstrate that this method closely approximates BPTT's gradient optimization, maintaining comparable performance while surpassing other efficient training techniques. This advancement is poised to enable more scalable and resource-efficient SNN training, particularly in environments with limited resources.", "questions": "1.\tCan the proposed rate-based backpropagation be implemented on existing neuromorphic chips with learning capabilities?\n2.\tLooking at the results in Fig.4, the impact of the number of timesteps on the time and memory looks constant. How have specific numbers of timesteps been selected for each dataset?", "limitations": "The limitations and societal impacts have been discussed in Appendix D.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe paper is very well written and documented.\n2.\tThe contributions have been discussed comprehensively.\n3.\tThe experiments have been conducted on multiple benchmarks.", "weaknesses": "Some important details (such as the top-level algorithm of the proposed rate-based backpropagation method and details of the experimental setup) are reported in the appendix, while, due to their importance, they should be moved to the main manuscript.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "#### **W1: Some important details (such as the top-level algorithm of the proposed rate-based backpropagation method anddetails of the experimental setup) are reported in the appendix, while, due to their importance, they should be movedto the main manuscript.**\n\nThank you for your suggestion. We appreciate your feedback and will consider moving more details to the main manuscript to facilitate a clearer understanding of the technical aspects for the readers.\n\n#### **Q1: Can the proposed rate-based backpropagation be implemented on existing neuromorphic chips with learning capabilities?**\n\nThis is indeed an interesting point. The proposed rate-based backpropagation highlights the significant aspect of training efficiency, which is very pertinent. To our knowledge, some neuromorphic chips [1,2] are already incorporating online learning schemes that perform backpropagation at specific timesteps while maintaining eligibility traces at the neural level. We would say that architectures supporting online training could easily adapt to support our proposed method. Computationally, our approach offers an effective optimization of both computational and memory complexity compared to BPTT. Thus, on neuromorphic chips that allow for custom configurations, such as Tianjic [3] and Loihi [4], rate-based backpropagation would be more practical than BPTT in terms of computational cost, storage demands, and communication overhead.\n\n\n#### **Q2: Looking at the results in Fig.4, the impact of the number of timesteps on the time and memory looks constant. How have specific numbers of timesteps been selected for each dataset?**\n\nThank you for your question. There are two aspects to consider in answering this. First and foremost, it's crucial to present our method as a fair comparison with recent works to determine whether the performance is compelling. Thus, we align the number of timesteps with those used in previous studies. Secondly, the number of timesteps used to train our SNN models indeed affects the inference latency, which is a key concern in the field of direct learning. Therefore, we often consider the tradeoff between higher inference accuracy and larger timesteps (which result in higher latency). The timesteps chosen for this study appear to be a good balance in benchmark scenarios.\n\n\n## Reference\n[1] Frenkel, Charlotte, and Giacomo Indiveri. \"ReckOn: A 28nm sub-mm2 task-agnostic spiking recurrent neural network processor enabling on-chip learning over second-long timescales.\" 2022 IEEE International Solid-State Circuits Conference (ISSCC). Vol. 65. IEEE, 2022. \n\n[2] Rostami, Amirhossein, et al. \"E-prop on SpiNNaker 2: Exploring online learning in spiking RNNs on neuromorphic hardware.\" Frontiers in Neuroscience 16 (2022): 1018006.\n\n[3] Pei, Jing, et al. \"Towards artificial general intelligence with hybrid Tianjic chip architecture.\" Nature 572.7767 (2019): 106-111.\n\n[4] Davies, Mike, et al. \"Loihi: A neuromorphic manycore processor with on-chip learning.\" Ieee Micro 38.1 (2018): 82-99.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Thank you to all the reviewers for your constructive comments and suggestions. We have addressed all the weaknesses and questions raised by the reviewers; detailed responses can be found in the corresponding sections of the rebuttal for each reviewer. In response to the reviewers' feedback, we have conducted additional experiments, including empirical validation on DVS-CIFAR10 and further experimental validation on performance and training costs, the results of those two are included in the global PDF.

Meta-review: This paper explores improved techniques for training spiking neural networks using backpropagation-through-time (BPTT). Specifically, the authors develop a technique relying on averaged rates-of-fire rather than detailed spiking dynamics. The authors show that this approach streamlines the computational graph, leading to less memory and time required for training. They demonstrate using various image categorization benchmarks that their approach leads to performance that is comparable to BPTT and better than existing efficient approaches for spiking networks.

The reviews for this paper were generally positive. Reviewers raised some concerns about novelty and clarity, which the authors sought to address in their rebuttal. Following this, the final average score was 6.0, and there was no reviewer that felt that this paper should be rejected. The AC concurred, and so, a decision to accept as a poster was reached.
Decision: Accept (poster)