Online Pseudo-Zeroth-Order Training of Neuromorphic Spiking Neural Networks
Authors: Mingqing Xiao
Qingyan Meng
Zongpeng Zhang
Di He
Dongsheng Li
Zhouchen Lin
Venue: ICLR 2026 Poster (2026)
Keywords: neuromorphic computing
spiking neural networks
non-backpropagation training
biological plausibility
pseudo-zeroth-order
Primary area: applications to neuroscience & cognitive science

Abstract: Brain-inspired neuromorphic computing with spiking neural networks (SNNs) is a promising energy-efficient computational approach. However, successfully training deep SNNs in a more biologically plausible and neuromorphic-hardware-friendly way is still challenging. Most recent methods leverage spatial and temporal backpropagation (BP), not adhering to neuromorphic properties. Despite the efforts of some online training methods, tackling spatial credit assignments by alternatives with competitive performance as spatial BP remains a significant problem. In this work, we propose a novel method, online pseudo-zeroth-order (OPZO) training. Our method only requires a single forward propagation with noise injection and direct top-down signals for spatial credit assignment, avoiding spatial BP's problem of symmetric weights and separate phases for layer-by-layer forward-backward propagation. OPZO solves the large variance problem of zeroth-order methods by the pseudo-zeroth-order formulation and momentum feedback connections, while having more guarantees than random feedback. Combining online training, OPZO can pave paths to on-chip SNN training. Experiments on neuromorphic and static datasets with both fully connected and convolutional networks demonstrate the effectiveness of OPZO with competitive performance compared with spatial BP, as well as estimated low training costs.

Published as a conference paper at ICLR 2026
ONLINE PSEUDO-ZEROTH-ORDER TRAINING OF NEU- ROMORPHIC SPIKING NEURAL NETWORKS
Mingqing Xiao 12 Qingyan Meng", Zongpeng Zhang", Di He Dongsheng Li',
Zhouchen Lin 2in 2,5x
Microsoft Research Asia
State Key Lab of General AI, School of Intelligence Science and Technology, Peking University
SPeng Cheng Laboratory
*Department of Biostatistics, School of Public Health, Peking University
6 Instiitte for Artificial Intelligence, Peking University
(mingqi ngxiao, dongsli i ) @mi crosoft. com, mengay@pcl.ac zongpeng. zhang98@gma i com, (dihe, zlin) @pku. edu.cn cn
ABSTRACT
Brain-inspired neuromorphic computing with spiking neural networks (SNNs) is promising energy-efficient computational approach. However, successfully training deep SNNs in a more biologically plausible and neuromorphic-harde friendly way is still challenging. Most recent methods leverage spatial and temporal backpropagation (BP), not adhering to neuromorphic properties. Despite the efforts of some online training methods, tackling spatial credit assignments by alternatives with competitive performance as spatial BP remains significant problem. In this work, we propose a novel method, online pseudo-zeroth-one Our method only requires single forward propagation with noise injection and direct top-down signals for spatial credit assignment, avoiding spatial BP's problem of symmetric weights and separate phases for layer-by-layee propagation. OPZO solves the large variance problem of zeroth-order methods by the pseudo-zeroth-orde formulation and momentum feedback connections, while having more guarantees than random feedback. Combining online training, OPZO can pave paths to on-chip SNN training. Experiments neuromorphic and static datasets with both fully connected and convolutional networks demonstrate the effectiivnees of OPZO with competitive performance compared with spatial BP, well as estimated low training costs.
(OPZO) training.
harward-backwand
INTRODUCTION
Neuromorphic computing with biologically inspired spiking neural networks (SNNs) is an energy-
efficient computational framework with increasing attention recently (Roy al.) 2019 Schuman et al. 2022). Imitating biological neurons to transmit spike trains for sparse event-driven computation as well as parallel in in-memory computation, efficient neuromorphic hardware developed, supporting
SNNs with low energy consumption (Davies et al.) 2018 Pei et al.) 2019 Wožniak et al. 2020 Rao
al.
2022
Davies 2021).
Nevertheless, supervised training of SNNs is challenging considering neuromorphic properties. While popular surrogate gradient methods can deal with the non-differentiable problem of discrete spikes (Shrestha & Orchard 2018 Wu et al. 2018 Nefici et al. 2019), they rely on backpropagation (BP) through time and across layers for temporal and spatial credit assignment, which biologically problematic and would be inefficient on hardware.
Particularly, spatial BP suffers from problems of weight transport and separate stages with update locking (Crick) 1989 Frenkel et al. 2021), and temporal BP is further infeasible for spiking neurons with the online property (Bellec et al., 2020). Considering learning biological systems with unidirectional local synapses, maintaining reciprocal forward-backkwar connections
backward
ccrrespoondig author

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with symmetric weights and separate phases of signal propagation is often viewed as biologically problematic (Nokland, 2016), and also poses challenges for efficient on-chip training of SNNs. Methods with only forward passes, with direct top-down feedback signals acting modulation in
biological three-factor rules (Frémaux & Gerstner 2016 Roelfsema & Holtmaat 2018). are more
efficient and plausible, e.g., on neuromorphie hardware (Davies 2021).
Some previous works explore alternatives for temporal and spatial credit assignment. To deal with temporal BP, online training methods are developed for SNNs (Bellee et al.) 2020 Xiao et al. (2022) With tracked eligibility traces, they decouple temporal dependency and support forward-in-time learning. However, alternatives to spatial BP still require deeper investigations. Most existing works mainly rely on random feedback (Nolland [2016] [ellecc et al.] (020), with limited guarantees and poorer performance than spatial BP. Some works explore forward gradients (Silver et al., 2022 Baydin et al. 2022), but they require an additional stage of heterogeneous signal propagation and perform poorly due to the large variance. Recently, |Malladi et al. (2023) show that zeroth-order (ZO) optimization with simultancous perturbation stochastic approximation (SPSA) can effectively fine-tune pre-trained large language models, but it requires specially designed settings, not suitable for general neural network training due the large variance. On the other hand, local learning has been studied, e.g., with local readout layers (Kaiser et al. (020) or
self-supervised learning (Hinton] 2022 Ororbia 2023). It complementary global learning and can improve some methods (Ren et 2003). As crucial component of machine learning, efficient global learning alternatives with competitive performance remain an important problem.
In this work, we propose a novel online pscudo-zeroth-order (OPZO) training method with only single forward propagation and direct top-down feedback for global learning. We first propose a pseudo-zeroth-order formulation for neural network training, which decouples the model and loss function and maintains the zeroth-order formulation for neural networks while leveraging the available first-order property of the loss function for more informative feedback error signals. Then we propose momentum feedback connections to directly propagate feedback signals to hidden layers. The connections are updated based on the one-point zeroth-order estimation of the expectation of the Jacobian, with which the large variance of zeroth-order methods can be solved, and more guarantees maintained compared with random feedback. OPZO only requires noise injection in the common forward propagation, flexibly applicable to black-box or non-differentiable models. Built upon online training, OPZO enables training in similar form the three- three-factor Hebbian learning based on direct top-down modulations, paving paths to on-chip training of SNNs. Our contributions include:
1. We propose pseudo-zeroth-oriee formulation that decouples the model and loss function for neural network training, which enables more informative feedback signals while keeping the zeroth-order formulation of the (black-box) model.
2. We propose the OPZO training method with a single forward propagation and momentum feedback connections, solving the large variance of zeroth-ordes methods and keeping low costs. Built on online training, OPZO provides a more biologically plausible method friendly for potential on-chip training of SNNs.
3. We conduct extensive experiments on neuromorphic and static datasets with fully connected and convolutional networks, as well as on ImageNet with larger networks fine-tuned under noise. Results show the effectiveness of OPZO in reaching competitive performance compared with spatial BP and its robustness under different noise injections. OPZO is also estimated have lower computational costs than BP potential neuromorphic hardware.
2 RELATED WORK
SNN Training Methods A mainstream method is spatio-temporal BP combined with surrogate gradient (SG) (Shrestha & Orchard 2018 Wu et al., 2018 Neftci et al. 2019), with many efforts on architecture or objective design Yao et al. 2024,) Xiao et 2024b Lv et al. [024|] Deng et al.)
2023, Guo et al. 20244 Xing et al. (2025). Another direction is to derive closed-form transformations or implicit equilibriums between encodings of spike trains (weighted firing rates or the first time to spike), and convert artificial neural networks (ANNs) to SNNs (Rueckauer et al.) 2017]] Deng & Gu 2021 Stöökkl& & Maass 2021 Meng et al. 2022b) directly train SNNs with gradients from the transformations (Zhou et al. 2021 Wu et al. 2021 Meng et al.) (202a)) or equilibriums (Xiao et al.,
2021
Martin et al. 2021 Xiao et al. 2023). To tackle the problem of temporal BP, some online
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training methods are proposed (Bellec et al. 2020 Xiao et al. 2022 Bohnstingl et al. 2022 Meng
al. 2023] Yin et al. (2023) for forward- in time learning, but many of them still require spatial BP.
Considering alternatives to spatial BP, Neftci et al. 2017 Lee et al. (2000) Bellec et al. (2020) apply
random feedback, Kaiser et al. (2020) propose online local learning, and Yang et al. (2022) propose
local tandem learning with ANN teachers. Different from them, propose a new global learning method with similar performance as spatial BP. Li et al. (2021) and Mukhoty et al. (2023) study zeroth-order properties for each parameter or neuron adjust surrogate functions leverage local zeroth-ordes order estimator for the Heaviside step function, lying in the Differently, in this work, zeroth-orde training refers to simultaneous perturbation for global network training without BP.
BP framework.
Alternatives to Spatial Backpropagation For more biologically plausible global learning, al- ternatives spatial BP are proposed. Target propagation (Lee et al. 2015), feedback alignment (FA) (Lillicrap et al. 2016), and sign symmetry (Liao et al. [016]] Xiao et al. 2018) avoid the weight symmetry problem by propagating targets or using random / only sign-shared backward weights, [Akruut et al. 2019) improves FA by learning it to be symmetric with forward weights. They, however, still need additional stage of sequential layer-by-layer backward propagation. Direct feedback alignment (DFA) [Nokland 2016 Launay et al. (020) improves FA to directly propagate errors from the last layer to hidden ones. However, random feedbacks have limited guarantees and perform much worse than BP. Some recent works study forward gradients (Silver et al. 2022 Baydin al. 2022, Ren al., 2023) Xiao al.) 2024a| [Baco & Chu 2024), but they require an additional heterogeneous signal propagation stage. suffering from biological plausibility issues and larger costs. There are also methods focusing on energy functions (Scellier & Bengio| 2017) or lifted proximal formulation (Li et al. 2020). Besides global supervision, some works turn to local learning, using local readout layers (Kaiser et al.] (020), forward-forwan contrastive learning (Hinton (2022), or Hebbian learning (Journé al. 2023). This work mainly focuses global learning and can be combined with local learning.
Zeroth-Order Optimization ZO optimization has been widely studied in machine learning, such as for black-box optimization (Grill et al. 2015), adversarial attacks (Chen et al., 2017), reinforcement learning (Salimans et al.) (2077). etc., at relatively small scales, but its application to direct neural network training is limited due to the variance caused by large number of parameters. Recently, Yue et al. ) theoretically shows that the complexity of ZO optimization can exhibit weak dependencies on dimensionality considering the effective dimension, and Malladi et al. (2023) proposes zeroth-order SPSA for memory-efficien fine-tuning pre-trained large language models with similar theoretical basis. However, it depends specially designed settings (e.g., fine-tuning under the prompt setting (Malladi et al. 2023) Gautam et al.) 2024)) which are not applicable to general neural network training, and requires two forward passes. Piang et al. (2024) proposes likelihood ratio method to train neural networks, but it requires multiple forward propagation proportional to the layer number in practice. Chen et al. (2024) considers the finite difference for each parameter rather than simultaneous perturbation and proposes pruning methods for improvement, limited in computational complexity. Differently, we propose a method for neural network training from scratch with only one forward pass for low costs and comparable performance to spatial BP.
3 PRELIMINARIES
3.1 SPIKING NEURAL NETWORKS
Imitating biological neurons, each spiking neuron keeps a membrane potential u, integrates input spike trains, and generates spike for information transmission once exceeds a threshold. is reset to the resting potential after a spike. We consider the commonly used leaky integrate and fire (LIF) model with the dynamics of the membrane potential as: Tm =-uu- Urrsst + R- [(t), for with input current I, threshold Vi., resistance R, and time constant Tm. When u reaches VA at time t the neuron generates spike and resets u to zero. The output spike train is s(t) Σnr 5(t - -f)).
Vih.
SNNS consist of connected spiking neurons. We consider the simple current model ((t)) = E, Wiyss(() + by, where i, represent the neuron index, wig is the weight and by is bias. The
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discrete computational form is:
Us [t ======pp X(u[() Vins:(1)) Σ Wiyss
(I)
+ 1|= H(, [t + V.n)
Here (x) the Heaviside step function, s,[0] is the spike signal at discrete time step i, and A < 1 is a leaky term (taken 1 = For multi-layer networks, use s to represent
the + 1) layer's response after receiving signals s'[t] from the I-th layer, i.c. the expression is u [t 1] A(u [t] Vast [t]) + s's 1] + b'.
Online Training of SNNs We build the proposed OPZO on online training methods for forward inn- time learning. Here online training refers to online through the time dimension of SNNs (Bellec et al. Xiao et al. 2022). as opposed to backpropagation through time, We consider OTTT (iiao
2020:
al.
2022 to online calculate gradients at each time by the tracked presynaptic trace à' f
Erse s s [r] and instantaneous gradient *, [4] =
N
8x"-11 06'+1
8x
as
Ww = gu [t]a(t]] [t] In OTTT, the instantaneous gradient requires layer-by-layer spatial ds'[t]
BP with surrogate derivatives for au'({] The proposed OPZO, on the other hand, leverages only
one forward propagation across layers and direct feedback to estimate Butta
[t] without spatial BP
combining surrogate gradients.
3.2 ZEROTH-ORDER OPTIMIZATION
Zeroth-order optimization is gradient-free method using only function values. A classical ZO gradient estimator is SPSA (Spal 1992). which estimates the gradient of parameters 0 for (()) on random direction z as:
20 L(0) C(O az) C(e az)
VL(O),
2a
where z is a multivariate variable with zero mean and unit variance, e.g., following the multivariate Gaussian distribution, and a a perturbation scale. Alternatively, we can use the one-sided formula- tion for this directional gradient (10+ax) L((0) z. These two-point estimations are unbiased estimator of V (O) in the limit a - 0 under the common assumptions of L-smoothness of L(O) and i.i.d. components of z with zero mean and unit variance (Nesterov & Spokoiny 2017: Duchi et al., 2015).
Considering biological plausibility and efficiency, estimation with a single forward pass more appealing. Actually, we can leverage the single-point zeroth-order estimation (ZOsp):
20.., (( C(B oz) a z.
For non-zero a in practice, it has the same expectation as the two-point method, Additionally, when is uniformly sampled from the unit sphere, the single-point estimation is an unbiased estimator of the smooth version of C: La (x) Ezes- [C(0 + az)), which does not require L to be differentiable (Flaxman et al. 2005).
The above formulation only requires a noise injection the forward propagation, and the gradients can be estimated with a top-down feedback signal, as shown in Fig. d). This is also similar to REINFORCE (Williams) 1992) and Evolution Strategies (Salimans et al. (017) in reinforcement learning, and is considered to be biologically plausible (Fiete & Seung] 2006). It is believed that the brain is likely to employ perturbation methods for some kinds of learning (Lllierap et al. 2020).
However, zeroth-order methods usually suffer from large variance, since two-point methods only estimate gradients a random direction and the one-point formulation has even larger variances. Therefore they hardly work for general neural network training. In the following, we propose momentum- based pseudo-20 pseudo-zeroth-ore zeroth-ortrr method to solve the problem, also only based on one forward propagation with noise injection and top-down feedback signals.
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(a) Online training of SNNs
(b) Spatial BP
(c) DFA
(d) 2O
(e) OPZO
W
W
WN-H
W
Random, jr
M!
W
W1
Wi
W
W
W
Time
Time
Spatial
Spacial bockward
Figure I: Illustration different training methods. (a) Online training of SNNs with tracked traces for temporal credit assignment. (b-e) Different spatial credit assignment methods. (b) Spatial BP propagates errors layer-by-layer with symmetric weights. (c) DFA directly propagates error signals from the top layer to the middle ones with fixed random connections. (d) single-point zeroth-order methods add perturbation during forward propagation, and afterward, the loss signal is passed to the middle layers. (e) The proposed OPZO method leverages momentum feedback connections based on perturbation vectors to directly propagate top-down error signals to neurons.
4 ONLINE PSEUDO HHEUD--ZEROTH-ORDER TRAINING
In this section, we introduce the proposed online pseudo-zeroth-order method. We first introduce the pseudo-zeroth-orfe formulation for neural network training in Section 4.1 Then in Section 4.2] introduce momentum feedback connections for еттоr propagation with -rothh-odder estimation of the model. In Section 4 we demonstrate the combination with online training and a similar form as the three-factor Hebbian learning. Finally, we introduce additional details in Section 4.4
4.1 PSEUDO-ZEROTH- ORDER FORMULATION
Since zeroth-order methods suffer from large variances, natural thought is to reduce the variance. However, ZO methods only rely on a scalar feedhack signal to act on the random direction z, making it hard to improve gradient estimation. To this end, introduce pseudorzeroth-order formulation. As we build our work on online training, we first focus on the condition of a single SNN time step.
Specifically, decouple the model function {(: 0) and the loss function L(-). For each input x, the model outputs o f(x; 0), and then the loss is calculated as ((o,yx) where yx is the label for the input. Different from ZO methods that only leverage the function value of L o f. we assume that the gradient of C(-) can be easily calculated, while keeping the zeroth-order formulation for f(;;0) 0). This consistent with real settings where gradients of the loss function have easy closed-form formulation, e.g., for Leaan-squrr--nne the softmax function a, VoC(o, yx) (o) Yx. while gradients of f(; f) are hard to compute due to biological plausibility issues or non-differentiabiiit of spikes.
(MSE) loss, Vx) = Yx. and for cross-entropy
(CE) loss with
With this formulation, we can consider feedback (error) signals V.L(o, yx) that carries more information than a single value of Lo fxx), potentially encouraging techniques for variance reduction. In the following, we introduce momentum feedback connections to directly propagate feedback signals to hidden layers for gradient estimation.
4.2 MOMENTUM FEEDBACK CONNECTIONS
We motivate our method by first considering the directional gradient by the two-point estimation in Section With decoupled f(; 0) and L(-) as in the pseudo-zeroth-ordeer formulation and Taylor expansion of C(-), the one-sided formulation turns into:
400 L (VoL(o,)x,, 0)
Ao VoL(o.
a
where o - f(x; f(x; 0 az), and Ao = o- This can be viewed as propagating the error signal with connection weight A To reduce the variance introduced by the random direction
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introduce momentum feedback connections across different iterations and propagate errors
Ao a
M = AM* + A)z
(5)
4 L M Vo
where M is initialized zero and &o denotes the iteration number. The momentum feedback connections can take advantage of different sampled directions z, largely alleviating the variance caused by random directions.
The above formulation only considers the directional gradient with two-point estimation, while we are more interested in methods with single forward pass. Actually, م be viewed as random estimator of EEx J (x) where (x) is the Jacobian of f evaluated x, and M can be viewed as approximating it with moving average. Therefore, we can similarly use one-point method:
M == AM + X)2 a
(6)
where z - is also an estimator of Ixx (x) with the same expectation as z _ when a is given in practice. also an unbiased estimator of Jacobian of the smoothed version of f, not requiring f to be differentiable (see Appendix A..] for details).
This leads to our method as shown in Fig. [e). During forward propagation, a random noise az is injected for each layer, and momentum feedback connections are updated based on and the model output o (information from pre- and post-synaptic neurons). Then errors are propagated through the connections to each layer. We consider node perturbation which is superior to weight perturbation" (Liliccrap et al. 2020)), then it has a similar form as the popular DFA (Nøkland 2016), while feedback weight a random matrix but the estimated Jacobian (Fig. [e,e)).
Then we analyze some properties of momentum feedback connections. We assume that M can converge to the estimated Ex JJ (x)
up to small errors and we focus on gradient estimation
with M Ex [ - e. We show that largely reduces the variance of the zeroth-order method (the proof and discussions are in Appendix A
Proposition 4.1. Let d denote the dimension of 0, m denote the dimension of (m < d), B denote the mini-batch size, B = Var [=]] Vo = -, Var [(V (Vofx)) = I, [(Voxx) where Lx the sample lossfor impul x, and VoLx and VoLx are the sample gradient for 0 and o, respectively. We further assume that the small error has i.i.d. components with zero mean and variance Ve, and
Vv.
Var [[x] EE [Lx]² V. 1 E. Var (Voox))),
Ti
let Vo,M - 2 Var [[V (xx)x (Bx JJ )ag. Then the average variance of the single-point zeroth-order method is: B ((d+ V) (d + 8 1)So V + SL) + +((a), while that of the pseudo-zeroth- method is: 1/ (mVVVo + mV.S. Vo.M).
Remark 4.2. Vo corresponds to the sample variance of spatial BP, and Vo.M would at similar scale as Vo (see discussions in (ppendix[A). Since V. is expected to be very small, the results show that the zeroth-order estimation has least times larger variance than BP, while the method can significantly reduce the variance, which is also verified in experiments.
Besides the variance, another question is that momentum connections would take the expectation of the Jacobian over data x, which can introduce bias into the gradient estimation. This due to the data-dependent non-linearity that leads to data-dependent Jacobian, which can be shared problem for direct error feedback methods without layer-by-layer spatial BP. Despite the bias, we show that
For = ó (W'x'). node perturbation estimates gradients for x ++1 and calculates gradients as V V (Van1f (W'x')) x' which has smaller variance than directly estimating gradients for weights. 'If the parameters of the model are tixed. M is approximating static matrix with projection to different directions, which can converge quickly. For the gradually evolving parameters, the expectation of the Jacobian over all samples may change slowly, and we also expect M to track this expectation slow time scale.

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under certain conditions, the estimated gradient can still provide descent direction (the proof and discussions are in Appendix (A).
Proposition 4.3. Suppose that J (x) is Li-Liptschitz continuous and e(x) is Le -iptsscchtt con- tinuous, x; is uniformly distributed, when [xx. (xx))()) > JLIL.Ax + e where Ax = x,| x|]² and e = ||EEEx, [e(x))]l, we have ( (Ex, J J (xx))(())) ,xx,, [Me(xi)]) O.
Xx,x, ||x:
Note that our analysis also holds for non-differentiable spiking neural networks. The single-poin estimation is actually an unbiased estimator for the smoothed fa with expectation over noise injection (Appendix [..I]. where f can be non-differentiable. This is similar to the stochastic setting where spiking neurons can be differentiable and gradients can be defined (see Appendix B for details). By treating f in the analysis as fa. the analysis effective.
4.3 ONLINE
SENEOO-ZERRTT--ORRE TRAINING
We build the above pseudo-zeroth-orre approach on online training methods to deal with spatial and temporal credit assignments. As introduced in Section 3.1] we consider OTTT (Xiao et al.] 2022] and replace its backpropagated instantaneous gradient with our estimated gradient based on direct top-down feedback. Then the update for synaptic weights has a similar form as the three-factor Hebbian learning (Frémaux & Gerstner 2016), and the global modulator is a direct top-down signal without sayer-by-lyeer BP:
AW,, oc
(-9)),
([((u(() where Wi, is the weight from neuron i to j. à, is the presynaptic activity trace, eu (t]) is a local surrogate derivative for the change rate of the postsynaptic activity (Xiao al. 2022). and of is the global top-down error (gradient) modulator. Here we leverage the local surrogate derivative because it can be well-defined under the stochastic setting (see Appendix [ and better fits the biological rule.
For potentially asynchronous neuromorphic computing, there may be a delay in the propagation of error signals. Xiao et al. (2022) show that with convergent inputs and certain surrogate derivatives, the gradient is still theoretically effective under the delay At, i.e., the update is based on āz(t At At)b(ugE [t At))g), Alternatively, more eligibility traces be used to store the local information, e.g., à, (4)),, and induce weight updates when the top-down signal arrives (Bellec et al., 2020). Our method shares these properties and we do not model delays in experiments for efficiency.
Moreover, the direct error propagations to different layers well the update of feedback con- nections in our method can be parallel, which can better take advantage of parallel neuromorphic computing than layer-by-layee spatial BP.
4.4 ADDITIONAL DETAILS
Combination with Local Learning There can be both global and local signals for learning in biological systems, and local learning (LL) can improve global learning approximation methods (Ren) et al. 2023). Our proposed method can be combined with LL as well. We consider introducing local readout layers, where fully connected readout is added for each layer with supervised loss. Additionally, we can also introduce intermediate global learning (IGL) that propagates global signals from a middle layer to previous ones with OPZO. More details can be found in Appendix C
About Noise Injection By default, we sample z from the Gaussian distribution. As sampling from the Gaussian distribution may pose computational requirements for hardware, we can also consider easier distributions such as the Rademacher distribution, which takes 1 and -1 both with the probability 0.5. Sampling from unit spheres is also feasible. Additionally, z is by default added the neural activities for gradient estimation based on node perturbation. To further prevent perturbation from interfering with sparse spike-driven forward propagation, we may empirically change the noise injection as perturbation before neurons (i.e., perturb on membrane potentials). while maintaining local surrogate derivatives for the spiking function. We will show in experiments that OPZO is robust to these noise injection settings. Additionally, we can leverage antithetic z, i.c., z and -z, for every two time steps of SNNs to further reduce the variance. More details can be found in Appendix C
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Table Accuracy (%) of different spatial credit assignment methods with online training on various datasets.
Method
N-MNIST
MNIST
CIFAR-10
CIFAR-100
BP (Xiao et al. BP (W/
2022
98.15-0.055
95.72:10.33
75.43:10.39
9..3810.022
9.000:1.006
64.82:10.09
(W/ LL)
95.72+0.71
76.07+0.41
89.82+0.16
64.88+0008
DFA (Nokland
2016
07.08
91.67+00.75
60.60+0.67
98.005+0.04
79.90 15 49.50==0.13
DFA (w/
91,43.10.59
61.77:0.62
/
82.38.10.22 54.76.10.21
DKP
Webster al. 2020
0.08
60.53.17.82
37.70.1..21
98.15.10.03
01.84
53.27..0.34
ZOip
72.90= 1.10
23.73+2238
31.67=0.24
86.53 11
49.04#00663
22.26=0055
OPZO OPZO (w/ LL)
9..2
94.33=00.16
0.8
98.34+0.100
85.74
60.9330.16
96.06.10.33
0.14
89.42.1.11
6..77
1.-01
1E406
0120
OP20
0P30 L3
1.6-05
variance
1F400
OP20
SPLI
B2 12
1.0.07
-P12
1.F46
B2 13
20 11
1E-11
12012
1.6122
20
0
10
20
30
40
200
250
20
epoch
epoch
MNIST
(b) DVS-CIFARIO
Figure 2: Results of gradient variances of OPZO, spatial BP, and zO on different datasets. "Li" denotes the i-th layer.
EXPERIMENTS
In this section, we conduct experiments on both neuromorphic and static datasets with fully connected (FC) and convolutional (Conv) neural networks to demonstrate the effectiveness of the proposed OPZO method. For N-MNIST and MNIST, we leverage FC networks with two hidden layers composed of 800 neurons, and for DVS-CIFARIO 10, DVS-Gesture CIFAR-10. 10. and CIFAR-100, leverage 5-layer convolutional networks. We will also consider a deeper 9-layer convolutional network, well fine-tuning ResNet-34 on ImageNet under noise. We take T = 30 time steps for N-MNIST, T 20 for DVS-Gesture, T = 10 for DVS-CIFARIO, and = 6 time steps for static datasets, following previous works (Xiao et al. 2022 Zhang & Li 2020 More training details can be found in Appendix
5.1 COMPARISON ON VARIOUS DATASETS
We first compare the proposed OPZO with other spatial credit assignment methods on various datasets in Table [ and all methods are based the online training method OTTT (Xaao et al. 2022) under the same settings. The compared methods include spatial BP, DFA (Nøkland 2016), DKP (Webster et al. 2020) that learns feedback connections in DFA, single-point zeroth-order method, and the combination with local learning. We do not consider local learning settings for FC networks since hidden layers. As shown the results, the zO method fails effectively optimize neural networks, while OPZO significantly improves the results, achieving performance at similar level as spatial BP. DFA with random feedback has large gap with spatial BP, especially on convolutional networks, while OPZO can achieve much better results. DKP improves DFA on static datasets, but it performs poorly on neuromorphic datasets and has significant gaps with OPZO on all datasets. When combined with local learning, OPZO (w/ LL) has about the same performance as BP (w/ L.L) and even outperforms BP (w/ L.L. on neuromorphic datasets. These results demonstrate the efffectiveness of OPZO for promising performance in more biologically plausible and neuromorphic sauromorphii-friendly approach, paving paths for direct on-chip training SNNs.
there are only
Note that our method is different line from most recent works with state-of-the-art performance (Li et al. 2023 Zhou et al. 2023 Guo et al.) 2024 Yao et al. 2024), which are based on spatio-tempone BP and focus architecture training objective improvement. We aim to develop alternatives to BP. focusing on more biologically plausible and hardware- hardware-friendly training algorithms. So we mainly compare different spatial credit assignment methods under the same settings.
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Table Accuracy (%) of different methods with a deeper network.
Table 2: Accuracy (%) of OPZO on CIFAR-I0 with different kinds of noise injection.
Method
DVS-Gesture
CIFAR-100
Distribution Pert. after neuron Pert. before neuron
Spatial BP
94.1011.02
65.96:10.52
Gaussian
85.73=+0.15
84.37+00.13
DFA (w/ LL) DFA (w/ LL&IGL)
93.40++0.49
52.94+0.20
Unit Sphere
86.01=0.228
84.50+00.13
93.29++0.33
54.17=00.54
Rademacher
85.69110.17
84.03:10.23
OPZO (w/ LL)
95.83-10.85
65.87.10.15
OPZO (w/ LL&IGL)
96.88=0028
66.13=00.15
5.2 GRADIENT VARIANCE
We analyze the gradient variance of different methods to verify that our method can effectively reduce variance for effective training. As shown in Fig. [ the variance of ZO.p is several orders larger than spatial BP, leading to the failure of effective training OPZO can largely reduce the variance have similar scale as BP, which is consistent with our theoretical analysis.
5.3 EFFECTIVENESS FOR DIFFERENT NOISE INJECTION
Then we verify the effectiveness of OPZO for different noise injection settings as introduced in Section 4.4 As shown in Table the results under different noise distributions and injection positions are similar, demonstrating the robustness of OPZO for different settings.
5.4 DERPER NETWORKS
We further consider deeper and larger networks. We first perform experiments with deeper 9-layer convolutional network. We leverage local learning and intermediate global learning (Section 4.4). As shown in Table 3 OPZO can also achieve similar performance or outperform spatial BP and significantly outperform DFA combined with these techniques. We also analyze pure OPZO without auxiliary techniques and its scalability to deeper networks with residual connections in Section D.5 showing that pure OPZO has more reliance on the proper network structure (residual connections) than BP for depth scalability.
We also conduct experiments for fine-tuning ResNet-34 on ImageNet under noise. This task is on the ground that there can be hardware mismatch, e.g., hardware noise, for deploying SNNs to neuromorphic hardware (Yang et al. 2022 Cramer et al. 2022), and we may expect direct on-chip fine-tuning to better deal with the problem. Our method is more plausible and efficient for on-chip learning than spatial BP and may be combined with other works aiming at high-performance training common devices in this scenario. We fine-tune pre-trained NF-ResNet-34 model released by Xiao et al. (2022) (original test accuracy 65. 15%) under the noise injection setting with different scales, As shown in Table OPZO can successfully fine-tune the model, while DFA and ZO fail. Spatial BP is ceuromorphhc-unfrinnde so its results are only for reference. The results show that OPZO can scale large-scale settings.
Table 5: Estimation of training costs on potential neuro- morphic hardware for N hidden-layer neural networks (n neurons for hidden layers, m neurons for the output, m & n). The costs focus on the error backward proce- dure. "** denotes parallelizable for different layers.
Table 4: Accuracy (%) of different meth- ods for fine-tuning ResNet-34 on Ima- geNet under different noise scale (n.s.). "Test"" refers to the direct test of the orig- inal model. "BP" refers to spatial BP.
Method
Memory
Operations
ImageNet
BP (if possible) O ((N 1)m² + mn) O ((N 1)n² + mm)
n.s.
Test
BP
DFA
ZO
OPZO
DFA*
0 ONNin)
0 (Nmn)
0.1
61.13
63.91
61.20
52.42
63.39
202*
O(Nn)
O(Nn)
0.15
54.01
62.13
54.59
30.32 60.96
OPZO*
O (Nmm)
(Nmn)

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5.5 TRAINING COSTS AND FIRING SPARSITY
Finally, we analyze and compare the computational costs of different methods. We mainly consider the estimated costs on potential neuromorphic hardware, which the target of SNNs. Since bio- logical systems leverage unidirectional local synapses, spatial BP (if assuming possible for weight transport and separate connections for error backpropagation. leading high memory and operation costs, as shown in 5 Differently, DFA and OPZO maintain direct top-down feedback with much smaller costs, which are also parallelizable for different layers. s0sp may have even lower costs by propagating only a scalar signal, but it is ineffective in practice. Also note that, different from previous zeroth-order methods that require multiple forward propagations, our method only needs one common forward propagation with noise injection and direct top-down feedback, keeping lower operation costs similar DFA. We also provide training costs on GPU in Appendix [D] and our method is comparable to spatial BP and DFA, since GPUs do not follow neuromorphic properties. It is interesting future work to consider applications to neuromorphic hardware that still under development (Davies) 2021 Schuman et al. 2022).
barkward stage) should maintain additional backward layer-by-laye
We further study the firing rate and synaptic operations of the trained models in Appendix [.2] showing that models trained by OPZO combined with local learning achieve the lowest operations, i.e., the most energy efficient. Additionally, we perform more analysis experiments of hyperparameters in Section D.4 Please refer to Appendix for more details and results.
6 CONCLUSION
In this work, we propose the online sseudo-zeroth-order method for training spiking neural networks more biologically plausible and herromoophiichhrdwar--ffe
way, with low estimated costs and competitive performance compared with spatial BP. OPZO performs spatial credit assignment by a single forward propagation with noise injection and direct top-down feedback with momentum feedback connections, avoiding drawbacks of spatial BP, solving the large variance problem of zeroth- order methods, and significantly outperforming random feedback methods. With online training, OPZO has similar form as three-factor Hebbian learning with direct top-down modulations, taking a step forward towards on-chip SNN training. Extensive experiments demonstrate the effectiveness and robustness of OPZO for both fully connected and convolutional networks neuromorphic and static datasets.
ACKNOWLEDGMENTS
Z. Lin was supported by the NSF China (No. 62276004).
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A DETAILED EXPLANATIONS AND PROOFS
In this section, we provide explanations and proofs for propositions in the main text.
A.1 EXPLANATION OF THE SMOOTHED FUNCTION
For the neural network function f with node perturbation given scale the smoothed version of defined fo (::0) 0) Eg [f 0, az)]. where f refers injecting noise for node perturbation. Similar to Flaxman et al. (2055) by extending the gradient to the Jacobian, we can show that the one-point formulation of z is an unbiased estimator of the Jacobian of fa.
Lemma A.1. When is uniformly sampled from the unit sphere, z is unbiased estimator of J. (x) given x, and further, are unbiased estimators of Fix J.. (x)
A.2 PROOF OF PROPOSITION
Proof. We first consider the average variance of the two-point 70 estimation V zz T Vo (a)). Since Var(æy) Var(z))ar(y)) + (yy) 2 + B(x)
for independent x and y.
and B[2 = Varla]] + E|2] = 1, for each element of the gradient under sample x, we have:
Var [(V)0 [x)
Var Σ 422 (VoLx), +0(a²)
Var
[2? (vxx)) + Σ
Var 222 (VoCx), (Pocx))+O((a) +0(a²)
j7i
Var [2]]] Var ((V [x Var [2]] E [V(x)) Var (VoLx) [
Σ
VVr [| Var (VoCx)y) + Var [ル=æ,] (VoCx), 1 Var (VoCx),) E Blazz))) +O(a2)
jtfi
(B + 1)Var 3E +
Σ
Var (Vo[x),) + E [(VoCx))]] +0a33)
J
d
= Var (Var(V)oo))lll (
Σ
Var [Vo(x),] + E [Vook>)]] + 0(07).
(8)
Taking the average of all elements, obtain the average variance for each sample (denoted as m Var):
m Var
40
Lx
d
1
Var
d
i=1
3
Var (VoLx), 3-1 Σ E (VVo
(Var (VoLx),] +E [VV(_a))) (
+
Σ ==1
+O(a²)
5.
1=1
(d + V + (d + 8 - 1) + Oa
99
For gradient calculation with batch size B, the sample variance can be reduced by B times, resulting in the average variance 1/ (dd 3oVo + (d + B 1)So) O(a²).
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V 20.0 L V/0 L f
Then derive the average variance of the single-point ZO estimation for each sample:
20p op Lx
maa
Lx = =aaar 70 Lx] + Var
2
1
3d+ (d+ 3- 1)50 + +O(a²)
+
Var Var [z+] + Var [cx]
B [zx] Var [z:] E [xx]
2
1 1
= -d+ + (d + 8 - 1)50 a²
V2 + 2 + ((a2)
(10)
For batch size B, the average variance is n ((d + V) O(a²).
+ V2 S2)
Next, we turn to the average variance of the pseudo-zeroth-orderr method 7.20g= = MV.6x
Ex J (x) E) VoCx. For each element, we have:
Var [(V/200 [x),
Var Σ (Ex (x](x)) 1), (Volx),
+ Var
Eas as (VoLx)))
(I)
j=l
m
FEx (x)()) i.j Var [Volx)g]
Σ j=l
m
VeVar (Volx),) + V.B (VooLx))'²)
Taking the average of all elements, we have the average variance for each sample:
m Var 1 Cx]
[J] (x)])) Var [Voox)))] + Σ V.Var (Vofx))] V.T [Vol=)))]
V.V. m.S. + Vo.M-
(12)
Then for batch size B, the average variance is (mVeVo Vo + mVeSo + VoM)..
Remark A.2. B Var [ = -(22)) = E(2{)) -1 depends on the distribution of 24. For the Gaussian distribution, ((2))) = and therefore 3 = 2. For the Rademacher distribution, EE22)) = 1 and therefore 0.
Remark A.3. The zero mean assumption on the small error e is reasonable, when we actually consider fa and z - is an unbiased estimator for Ex J (x) (Lemma A.I) the expectation of the error can be expected to be zero.
Remark A.4. Ve and V..M may not be directly compared considering the complex network function, we may make brief analysis under some simplifications. For (VoLx); = (J) Vx)xx) let
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Jz and V, denote (J](x)) and (VoCx), for short, we have
m Ji JR v
Var (VoCx)+] = Var
Σ Var JJ, vyl + Σ Cov
[Ja.
Vi., Vial
= -[[van Var [V [J Var زرال
E 1 1 2Cov (Ja.. V E
ECov
Cov [Ji.h Vi Je,z Vall.
Snija
(13)
If we ignore covariance terms and assume E [V } = 0, this is simplified to Σ, Var [V,] [J and then Vo is approximated as Eng Var [V,] E [Jβg]. which has a similar form as Vo,M
Ex Var ((Volx) (Ex [J] (xx) (xx) Das except that the second moment is considered. Under this condition, the scales of V..M and Vo may slightly differ considering the scale of elements of J (x), but overall, Vo,M would be at a similar scale as Vo compared with the variances of the zeroth-orde methods that are least d times larger which proportional to the number of intermediate neurons.
A.3 PROOF OF PROPOSITION
Proof. Since J. (x) is Li-Litsschitz continuous and e(x) is L. Le-Litsschitz continuous, we have (x,) - J/ (x,) that
s L, ||x, - x,11 |e((x)) -e(x,)l I Le ||x; xyll- Then with the equation Lij(ai- -aj)=-a))b;)) bb)) = E, abbi 2/2 Ee, azbj. have
Rx: [J] (xx)) Exx [Bxx, [J] (x,)] + exx:)]l
1 Je(x))e(x))
1 ET(xx))
1 Σ e(x)) elEx, [e(xi)]
Jr(x;) - JF(x)) (e(x:) e(x;)) ellxx. [e(x:)]
2n
X,
(J((x.) - -F((x)) [[(((x)) - c(xx))|| + |IEx, [c(x;)]1
(14)
2n² X,
< 2n
Lzbellx;
[e(x:)]l
X;
Lybin Ax +
< Ex [J]((xs)))))) (x)) ||.
Therefore,
(Ex,
[ (x)))(x;)) Bxx [Me(x;))) [J] (xx))((x ]][ - (IEx. [J] (x:)e(x;) Ex. [J] (x:))(()) Ex, [Me(x:)])
||xx,
Ex, JJ (x))()))) [xx [J] (x))(())) ||x [J] (xx))() - Ex, [Bxx [J] (x,)] + E) e(x:)] ||
0.
(15)
L
Remark A.5. Ly will depend on the smoothness of the network, for example, L, = for linear networks. This will influence the condition of effective descent direction considering the gradient scale as in the proposition. Note that these assumptions are not necessary premises, and we have verified the effectiveness of the method in experiments.
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B INTRODUCTION TO LOCAL SURROGATE DERIVATIVES UNDER THE STOCHASTIC SPIKING SETTING
In this section, we provide more introduction to the stochastic spiking setting, under which spiking neurons can be locally differentiable and there exist local surrogate derivatives.
Biological spiking neurons can be stochastic, where a neuron generates spikes following a Bernoulli distribution with the probability the c.d.f. of distribution w.r. 2 Via, indicating higher probability for a spike with larger ul{] Vnn- That is, si(4] is a random variable following a
1) F(u,[t] V++)- With reparameterization. this can be formulated as s,[t] = (u; [t] - Via- - 24) with random noise variable 2; that follows the distribution specified by F. Different F corresponds to different distributions and noises. For example, the sigmoid function corresponds to logistic noise, while the erf function corresponds derivatives can be introduced for the spiking function (Shekhovtsov & Yanush 2021 Ma et al. (023).
(0, 1) valued Bernoulli distribution with the probability of 1 as p(si(4
Gaussian noise. Under the stochastic setting, the local surrogate
Specifically, consider the objective function which should turn the expectation over random variables under the stochastic model. Considering one-hidden-lan with the input x connecting to n spiking neurons by the weight W and the neurons connecting to an output readout layer by the weight O. Different from deterministic models with the objective function Ex [C(s)], where s = Hu Via), u = Wx, under the stochastic setting, the objective is to minimize:
network with one time step,
Ex pL
(16)
For this objective, the model can be differentiable and gradients can be derived (Shekhovtsov & Ma et al. 2023). We focus on the gradients of u, which can be expressed as:
Yanush 2021
0
E. s [C(s)] δ
Σ
I ((s)
Ou
du
(17)
) (ou (ossss))) C(s).
ΣΣ
II p(se w)
Then consider derandomization to perform summation over 8; while keeping other random variables
fixed (Shekhovtsov & Yanush 2021). Let s-1 denote other variables except s,. Since s; is {0, 1} valued, given S-4, we have
Öp(s: W) C((s-), sil) ap(s, W)
du L(s) + 0(1 - p(s; ()) C(s(s)
Ou
du
(0.1).
(18)
ap(s. W)
(C(s) - L(su:))
du
where is a random sample considering s; (the RHS is invariant s;), and sμ denotes taking s; as the other state for s. Given that Σ., p(s: () = 1. Eq. (17 is equivalent to
du ((=)
p(s p(s (W) Kz
II
Σ
i Du
p(s; w)) ((ss
@p(s, W)
ΣΣ 8
II P+1
ppss://)
Σ p(s:/W) W)
(C(s) L(sw))
du
(19)
Op(s: W) (C(s) - L(s
Σ
Il p(s Σ
du
Öp(s,
E Es-n Σ
(C(s) L(s)))-
du
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Taking sample of in each forward procedure allows the unbiased gradient estimation as the Monte Carlo method. In this equation, considering the probability distribution, we have:
Öp(s. W) = F' V
(20)
du
where F" is the derivative of F, corresponding to a local surrogate gradient, c.g., the derivative of the sigmoid function, triangular function, etc.
The term ((s)
[(s corresponds to the error, and the above derivation is also similar to REIN- FORCE /Williams 1992). However, since it relies derandomization, simultaneous perturbation is infeasible in this formulation, and for efficient simultaneous calculation of all components, we may follow previous works (Shekhovtsov & Yanush (021) to tackle it by linear approximation:
[(s) - (ss)) & ac(x), enabling simultaneous calculation given gradient @C(s)
de This approxima-
de,
tion may introduce bias, while can be small for verrparameterized neural networks with weights at the scale of va..
where dn is the neuron number. This means that for the elements of the readout
Os, flipping the state of s, only has O va.) influence.
The deterministic model may be viewed as special case, e.g., with noise always zero, and Shekhovtsov & Yanush (2021) show that the gradients under the deterministic setting can provide a similar ascent direction under certain conditions. Also, the noise injection in our method is similar to introducing the randomness in stochastic neuron model.
Therefore, spiking neurons be differentiable under the stochastic setting and local surrogate derivatives can be well-defined, supporting our formulation as introduced in the main text. Our
pseudo-zerooe ceroth-order method approximates dcts) fitting the above formulation. Also note that the above derivation of surrogate derivatives is local for one hidden layer for multi-layer networks, while we may iteratively perform the above analysis to obtain the commonly used global surrogate gradients, there can be expanding errors through layer-by-laye propagation due to the linear approximation error. Differently, our OPZO performs direct error feedback, which may reduce such errors.
C MORE IMPLEMENTATION DETAILS
C.I LOCAL LEARNING
For experiments with local learning, we consider local supervision with a fully connected readout for each layer. Specifically, for the output s' of each layer, we calculate the local loss based on the readout r' R's' as C(r', y). Then the gradient for is calculated by the local loss and added to the global gradient based our OPZO method, which will update synaptic weights directly connected to the neurons. We assume the weight symmetry of local learning for propagating errors, i.e., using a feedback weight P R' propagate errors as pl ac(r',y) This is because for the single linear
Ør'
layer that directly connects to the output, the weight Pl can be learned to symmetric to R through symmetric local Hebbian-like update rule, i.c., both of them updated by (C(F' T based on dr =rr)) -y for CE loss). This mechanism does not require global error information and is compatible with the intended constraints, while it only applicable to the single (linear) layer condition. Kaiser et al. (2020) also show that fixed random matrix can be effective for such kind of local learning.
pre- and post-synaptic information (e.g., ac(r' rd y for MSE loss and 8C(r' 2r
Or
We also consider intermediate global learning (IGL) as a kind of local learning. That is, we choose middle layer to perform readout for loss calculation, just as the last layer, and its direct feedback signal will be propagated to previous layers. For the experiments with 9-layer network, we choose the middle layer the fourth convolutional layer.
C.2 NOISE INJECTION
For each time step of SNNs, we sample a z and add it to the network after or before the neural activities (see Section 4.4). Compared with the two-point zeroth-order estimation, the considered one-point method can have a much larger variance. To further reduce the variance, we can leverage antithetic i.e., and z, for every two time steps of SNNs. Since SNNs naturally have multiple
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time steps and the inputs for different time steps usually belong to the same object with similar distributions, this approach may roughly approximate the two-point formulation without additional costs.
C.3
TRAINING SETTINGS
C.3.1
DATASETS
We conduct experiments on N-MNIST (Orehard et al. 2013), DVS-Gesture (Amir et al.) 2017). DVS- CIFARIO (Li et al. 2017), MNIST (LeCun et al. 1998). CIFAR-10 and CIFAR- 100 (Krizhevsky & Hinton 2009). as well as ImageNet Deng et al. 2009).
N-MNIST N-MNIST a neuromorphic dataset converted from MNIST by a Dynamic Version Sensor (DVS), with the same number of training and testing samples MNIST. Each sample consists of spike trains triggered by the intensity change of pixels when DVS scans a static MNIST image. There are two channels corresponding to ON- and OFF-event spikes, and the pixel dimension is expanded to 34 x 34 due to the relative shift of images. Therefore, the size of the spike trains for each sample is 34 x 34 x 2 T, where T is the temporal length. The original data record 300ms with the resolution of μus, We follow Zhang & Li (2020) to reduce the time resolution by accumulating the spike train within every 3ms and use the first 30 time steps. The license of N-MNIST is the Creative Commons Attribution Sarrbution-ShareAAe
4.0 license.
DVS-Gesture DVS-Gesture is neuromorphic dataset recording 11 classes of hand gestures by DVS camera. It consists of 1,176 training samples and 288 testing samples. Following Fang et al. 202I), we pre-pon pre-osssss the data to integrate event data into 20 frames, and we reduce the spatial resolution 48 48 by interpolation. The license of DVS-Gestee is the Creative Commons Attribution 4.0 license.
DVS-CIFARIO DVS-CIFARIO is the neuromorphic dataset converted from CIFAR-10 10 by DVS, which is composed of 10,000 samples, one-sixth of the original CIFAR-10. It consists of spike trains with two channels corresponding to ON- and OFF-event spikes. We split the dataset into 9000 training samples and 1000 testing samples as the common practice, and we reduce the temporal resolution by accumulating the spike events (Fang et al.) (02T) into 10 time steps well as the spatial resolution into 48 48 by interpolation. We apply the random cropping augmentation similar CIFAR-10 to the input data and normalize the inputs based on the global mean and standard deviation of all time steps. The license of DVS- OVS-CIFARIO is CC BY 4.0.
MNIST MNIST consists of 10-class handwritten digits with 60,000 training samples and 10,000 testing samples. Each sample is 28 x 28 grayscale image. We normalize the inputs based on the global mean and standard deviation, and convert the pixel value into a real-valued input current at every time step. The license of MNIST is the MIT License.
CIFAR-10 CIFAR-IC 10 consists of 10-class color images of objects with 50,000 training samples and 10,000 testing samples. Each sample is a 32 x 32 x 3 color image. We normalize the inputs based the global mean and standard deviation, and apply random cropping, horizontal flipping, and cutout (DeVries & Taylor 2017) for data augmentation. The inputs to the first layer SNNs at each time step are directly the pixel values, which be viewed as real-valued input current.
CIFAR-100 CIFAR-100 is dataset similar to CIFAR-10 except that there 100 classes of objects. It also consists of 50,000 training samples and 10,000 testing samples. We use the same pre-processing CIFAR-10.
The license of CIFAR-10 and CIFAR-100 is the MIT License.
ImageNet ImageNet- IK a dataset of color images with 1,000 classes of objects, containing 1,281,167 training samples and 50,000 validation images. We adopt the common pre-possessing strategies to first randomly resize and crop the input image to 224 x 224, and then normalize after the random horizontal flipping data augmentation, while the testing images are first resized to
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256 x 256 and center-cropped to 224 x 224, and then normalized. The inputs are also converted to real-valued input current at each time step. The license of ImageNet is Custom (non- commeremer......
C.3.2 TRAINING DETAILS AND HYPERPARAMETERS
For SNN models, following the common practice, we leverage the accumulated membrane potential of the neurons at the last classification layer (which will not spike or reset) for classification, i.e., the classification during inference is based on the accumulated u [T] = >i= oft]. where oft] = W [t] bb which can be viewed as an output at each time step. The loss during training is calculated for each time step as [(o[t], y) following the instantaneous loss in online training with the loss function as a combination of gross-entropy (CE) loss and Seeun--suar-errre (MSE) loss (Xiao) ct al. 2022). For spiking neurons, Vo, and 0.5. We leverage the sigmoid-like like local surrogate (V w)/ez
derivative, i.e., (b( = a (1+=(V1a-)//o with @1 = 0.25. For convolutional networks, we apply the scaled weight standardization (Brock et al. 2021) as in Xiao et al. (2022).
For our OPZO method, as well as the ZO method in experiments, a is by default set as 0.2 initially and linearly decays to 0.01 through the epochs, in order to reduce the influence of stochasticness for forward propagation. In practice, this schedule is not critical (see analysis in Section D.4). For fine-tuning on ImageNet under noise, a is set as the noise scale, and we do not apply antithetic variables across time steps, in order to better fit the noisy test setting (perturbation noise is before the neuron). practice, we remove the factor 1 //v for the calculation of M, because in the single-point setting, the scale of Ö is larger than and not proportional to This only influences the estimated gradient with a scale a, and may be offset by the adaptive optimizer. Analysis experiments show that this factor is also not critical (see Section D..4). For gradient variance analysis, we keep this factor in order to have a comparable gradient scale.
For N-MNIST and MNIST, consider FC networks with two hidden layers composed of 800 neu- rons, and for DVS-CIFARIO, DVS- Gesture, CIFAR-10, and CIFAR-106 100, we consider 5-layer Cony networks
or 9-layer Conv networks under the 1-C----------1233
deeper network setting
FC). We train our models on common datasets by the AdamW optimizer with learning rate 2c-4 and weight decay 2e-4 (except for ZO, the learning rate is set as 2c-5 on DVS-CIFARIO, MNIST, CIFAR-10. and CIFAR-100 for better results). The batch size is set as 128 for most datasets and 16 for DVS-Gesture, and the learning rate is cosine annealing to O. For N-MNIST and MNIST, we train models by 50 epochs and we apply dropout with the rate 0.2 (except for ZO). For DVS-Gesture, DVS-CIFAR 10, CIFAR- 10, and CIFAR-100, we train models by 300 epochs. For DVS-CIFAR 10, we apply dropout with the rate 0.1 (except for ZO). We set the momentum coefficient for momentum feedback connections as x= = 0.99999 (except for DVS-Gesture, it is set X 0.999999 due to smaller batch size), and for the combination with local learning, the local loss is scaled by 0.01.
For fine-tuning ImageNet, the learning rate is set as 2e-6 (and 2 for ZO) without weight decay, and the batch size is set as 64. The perturbation noise is before the neuron, i.e., added to the results after convolutional operations. For BP, train 1 epoch. For DFA, ZO, and OPZO, we train 5 epochs. We observe that DFA and ZO fail after 1 epoch, so we only report the results after 1 epoch, and for OPZO, the results can continually improve, so we report the results after 5 epochs. The 1-epoch and 5 5-epoch results for OPZO are 63.04 and 63.39 under the noise scale of 0. and 59.50 and 60.96 under the noise scale of 0.15.
The code implementation is based on the PyTorch framework, and experiments are carried out on one NVIDIA GeForee RTX 3090 GPU (each experiment takes several hours). Experiments are based on 3 runs of experiments with the same random seeds 2022, O, and 1. Note that the results hardly change for more runs of experiments: for OPZO results on DVS-CIFAA 10 with the largest standard deviation, 10 runs have almost the same results (72.69 + 0.62 vs. 72.77 + 0.82).
For gradient variance experiments, the variances are calculated by the batch gradients in one epoch, ie., var - x where g; is the batch gradient, g is the average of batch gradients, and is the number of batches multiplied by the number of clements in the gradient vector.
n
DFA and DKP For DFA, the direct error feedback weight is randomly initialized following the Kaiming initialization strategy. DKP (Webster et al. 2020 is based on the formulation of DFA
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Table 6: Brief comparison of training costs on GPU for CIFAR-I0 with convolutional networks. i means manual implementation of spatial BP with layer-by-layer backpropagation, which is in similar fashion as other methods. means using automatic differentiation implemented by PyTorch with low-level code optimizations.
Method
Memory
Time per epoch
Spatial BP
2.8G 2.9G
49s /45s
DFA
2.8G
44s
70 OPZO
2.8G
46s
2.9G
46s
DFA (w/ LL) OPZO (w/ LL)
3.0G
50s
3.1G
51s
and updates feedback weights similar to Kolen-Pollack learning, which calculates gradients for feedback weights by the product of the middle layer's activation and the error from the top layer. The feedback weights are initialized as zero, and we treat them as parameters to be optimized by the Adam optimizer. Its basic thought trying to keep the update direction of feedback and feedforward weights the same, but it may lack sufficient theoretical grounding. As DKP is designed for ANN, we implement it for SNN with the adaptation of activations to pre-synaptic traces for feedback weight learning (similar to the update of feedforward weight). As shown the results, compared with DFA, DKP can have around 2-3% performance improvement on CIFAR-I0 10 and CIFAR- 100, which is similar to the improvement in its paper. However, DKP cannot work well for neuromorphic datasets. And OPZO significantly outperforms both DKP and DFA on all datasets.
D ADDITIONAL RESULTS
D.1 TRAINING COSTS ON GPU
We provide a brief comparison of memory and time costs of different methods GPU in Table 6 Our proposed OPZO has about the same costs as spatial BP and DFA. If we exclude some code-level optimization and implement all methods in similar fashion, DFA and OPZO are faster than spatial BP, which is consistent with the theoretical analysis of operation numbers. Note that this is only brief comparison, as we do not perform low-level code optimization for OPZO and DFA, for example, the direct feedback of OPZO and DFA to different layers can be parallel, and local learning for different layers can also theoretically be parallel, to further reduce the time. As described in the main text, the target of neuromorphic computing with SNNs would be potential neuromorphic hardware, and OPZO and DFA can have lower costs, while GPUs generally do not follow the properties. Since neuromorphic hardware is still under development and we have limited access, we mainly simulate the experiments on GPUs, and it can be future work to consider the combination with neuromorphic hardware implementation.
Also please note that these methods are all based on online training, so the memory costs (agnostic to time steps) are already largely reduced compared with BPTT (proportional to time steps) (Xiao et al. 2022).
D.2 FIRING RATE AND SYNAPTIC OPERATIONS
For event-driven SNNs, the energy costs on neuromorphic hardware are proportional to the spike count, or more precisely, synaptic operations induced by spikes. Therefore, we also compare the firing rate (i.e., average spike count per neuron per time step) and synaptic operations of the models trained by different methods. As shown in Table 7 on both DVS-CIFARIO I0 and CIFAR-1O, OPZO (w/ LL) achieves the lowest average total firing rate and synaptic operations, indicating the most energy efficiency. The results also demonstrate different spike patterns for models trained by different methods, and show that LL can significantly improve OPZO, while it can hardly improve DFA and spatial BP. It may indicate OPZO as a better, more biologically plausible global learning method to be combined with local learning.
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Table 7: The firing rate (fr) and synaptic operations (SynOp) induced by spikes for models trained by different methods on DVS-CIFARIO and CIFAR-10.
DVS-CIFAR10
Method
Layerl fr
Layer2 fr
Layer3 fr
Layer4 fr
Total
SynOp
Spatial BP
0.1763
0.1733
0.2394
0.3575
0.1904
1.42 x 10
Spatial BP (w/ LL)
0.2272
0.1618
0.2199
0.3439
0.2122
1.51 109
DFA
0,2693
0.4564
0.4783
0.4930
0.3574
2.91 109
DFA LL)
0.2433
0.4531
0.4848
0.4919
0.3430
2.84 x 10
OPZO
0.2435
0.3446
0,4212
0.4222
0.3021
2.41 10
OPZO (w/ LL)
0.0406
0.0614
0.1451
0.2838
0.0691
0.53 x 10
CIFAR-10
Method
Layer1 fr
Layer2 fr
Layer3 fr
Layer4 fr
Total fr
SynOp
Spatial BP
0.2005
0.1679
0.1067
0.0493
0,1734
0.76 x 109
Spatial BP (w/ LL)
0.1870
0.1474
0.0978
0.0470
0.1589
0.69 x 10
DFA
0.1769
0.3787
0.4314
0.4149
0.2759
1.40 x 10
DFA (w/ LL)
0.1196
0.3180
0.4089
0.3878
0.2235
1.16 x 109
OPZO
0.1563
0.2861
0.3496
0.2754
0.2229
1.12 x 10
OPZO (w/ LL)
0.0400
0.0670
0.1159
0.2157
0.0640
0.30 10
0.8
0.95
0.55
06
0.5
TPP
0.3
DEA
I1A
20
CR7DD
0.16
013
0.1
01
30
40
90
0
10
20
30
50
100
190
200
20
3D
50
100
150
200
250
30
(a) N-MNIST, FC, train
(b) N-MNIST, FC, test
(c) DVS-CIFAR10, Cony, train
(d) DVS-CIFAR1O, Conv, test
Figure Training dynamics of different methods on N-MNIST and DVS-CIFARIO.
D.3 TRAINING DYNAMICS AND GRADIENT SIMILARITY
We present the training dynamics of different methods in Fig. For the fully connected network on N-MNIST, OPZO achieves similar convergence speed as spatial BP, which is better than DFA and much better than ZO. For the convolutional network on than spatial BP while still performing much better than DFA and 7O, and when combined with local learning, OPZO (w/ LLL) achieves similar training convergence speed as spatial BP well as better testing performance.
OPZO itself is slower
We further present the cosine similarity between estimated gradients and backpropagated gradients with surrogate derivatives in Fig. The results show that the cosine similarity of different layers between OPZO and BP remains in the range 0.5-0.9 throughout training. whereas DFA and BP is typically below 0.1 for most layers. This indicates that the bias introduced by momentum feedback does not significantly distort the gradient direction compared to DFA, and the training can converge with effective descent directions.
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OPZO vs. BP
DFA vs. BP
Cosine similarity
0.5
Cosine similarity
0.8
0.5
0.6
Layer
0.4
Layer2
0.4
Layer 2
Layer
Layar 3
0.2
Layer
0.2
Layer 4
epoch
epoch
Figure 4: Cosine similarity between OPZO and BP with surrogate derivatives as well as between DFA and BP on CIFAR-100.
Table 8: Analysis results of different À on CIFAR-100.
À = 0 (w/o momentum)
=
À = 0.99
X = 0.999
À 0.9999
À 0.99999
16.08
39.05-10669
50.65:10.05
58.25
60.92.10.11
60.93
Table 9: Analysis results of updating M throughout training on CIFAR-100.
stop updating after 10 epochs
OPZO
57.17
60.93=0.16
Table 10: Analysis results of settings with initial perturbation scale a 0.2 on CIFAR-100,
w/ schedule
w/o schedule
w/o schedule and w/ 1 factor
00.93
61.85-10.09
61.65+10.40
Table 11: Analysis results of different perturbation scale a without scheduling on CIFAR-100.
1.
0.2
a 0.02
a 0.002
58.34
61.85=0.099
58.33
49.97
D.4 ANALYSIS OF HYPERPARAMETERS
As shown in the results, OPZO requires a large our scenario. This is due the large variance of single-point zeroth-order approximation, so we require relatively small 1 - À for smoothing M to approximate the expectation Ex J (x) (more precisely, Ex JJ (x)) ). A smaller cannot properly deal with the large variance, leading to inferior performance. While is large, this does not mean M is quasi-static, because the objective of the expectation of Jacobian is slowly changing throughout training. To validate this in experiments, we stop updating M after 10 epochs, and the performance drops as shown in Table g
We first study the influence of the momentum coefficient À in Table g
We then analyze the influence of the perturbation scale a. We first evaluate the effect of the scheduling of a and removing the factor 1 (Section C.3.2). As shown in Table 10. the scheduling has slightly negative influence on the performance while the factor has negligible impact. We further analyze different perturbation scales without scheduling in Table 11 As shown in the results, the scale around 0.2 works best. This is likely due to the property of spiking neural networks, where we set the spiking threshold as 1: if the perturbation scale is too small, the perturbation hardly influences the spiking generation, leading to imprecise estimation. Therefore, for spiking neural networks, a good scale choice would be fixed 0.2 throughout the training.
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Table 12: Analysis results of deeper networks without auxilary techniques on CIFAR-100.
Network structure
Spatial BP
DFA
%0
OPZO
5-layer network
64.82=0.09
49.50=1..3
22.26-10.51
60.93:1.116
9-layer network
65.96=0.52
48, 08,14
4.85+1.21 /
56.89-10.7
9-layer network (w/ residual)
69.5
48.5
63,4
18-layer residual network (smaller channels)
66.78
46.83
/
61.86
34-layer residual network (smaller channels)
67.36
45.06
/
61.17
D.5 ANALYSIS OF PURE OPZO FOR TRAINING DEEPER NETWORKS FROM SCRATCH
In this section, we provide more analysis on the pure OPZO for training deeper networks from scratch without auxiliary techniques. as shown in Table [2]
First, for the plain 9-layer network, OPZO has degraded performance compared with 5-layer network, but still significantly outperforming DFA and ZO. This related our theoretical analysis of the bias of average Jacobian, which shows that the smoothness will influence the condition of effective descent direction (Remark (..5). The plain deeper networks' unsmoothness tightens the condition, leading to inferior performance. So some auxiliary mechansims may be required to alleviate the problem.
Then, we further show that residual connections can largely alleviate the problem, in consistent with our theoritical analysis and experiments that pure OPZO can effectively fine-tune ResNet-34 on ImageNet. As shown in Table 12 with residual connections, OPZO does not degrade as depth grows and can scale to 34-layer networks in the train from scratch setting as well, while DFA still has degraded performance. This is because residual connections smoother, so the problem is largely alleviated. While there is still some performance gap with BP, OPZO significantly outperforms DFA and can be further enhanced with auxiliary techniques.
make the network function
Therefore, pure OPZO also has the ability to train deeper networks from scratch, while it has more reliance on the proper network structure (residual connections) than BP.
Additionally, we report results for training a NF-ResNet18 model (T=4) from scracth on ImageNet for 100 epochs using the Adam optimizer (Ir=le 4). The performance of pure OPZO, DFA, and BP is 19.0%, 5.6%, and 52.9%, respectively. The results show that while OPZO significantly outperforms DFA, it still has much room for improvement compared with BP in this large-scale offline training, likely due to a more non-smooth optimization landscape as indicated by our theoretical analysis. This suggests that OPZO may require additional techniques such as local learning for this scage training- wwwo--rroottttte scenario. However, we emphasize that OPZO is not intended to replace BP for high-compute offline training, which orthogonal to our neuromorphic objective. Our goal is hardware-friendl. on-chip learning for neuromorphic SNNs, where starting from scratch is rarely necessary (similar to our brains that adapt rather than relearn entirely), and post-deployment adaptation and continual learning are key scenarios. Our ImageNet fine-tuning experiments show that pure OPZO can scale to large networks without auxiliary techniques, validating its effectiveness in this setting. This demonstrates OPZO's complementary role offline BP methods.
E
MORE DISCUSSIONS
E. LIMITATIONS
This paper mainly focuses theoretical groundings and simulation experiments with GPUs for the proposed method, while no implementation on neuromorphic hardware is included due to our limited access to it. Future work can consider the implementation those hardware with more engineering efforts, e.g., on Loihi2 (Davies] 2021) that can support three-factor learning rules as described in their technical report.
As discussed Section 5 our method a different line from many recent SNN works with state- of-the-art performance, focusing on more biologically plausible and hareware-friendly training
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Table 13: Results of different methods without truncating temporal gradients on DVS- CIF.CI.ARIO.
Network structure
STBP
DFA
PZO
5-layer CNN (sWS)
76.17:00221
60.00++.22
73,20:10.08
5-layer CNN (BN)
77.23+00.46
60.77110.9
74.13::0.52
algorithms. So we mainly evaluate the effectivvness of the method with comparisons under various settings, not pursuing the state-of-the-ar performance. On the other hand, as discussed in the experiments of fine-tuning ResNet-34 on ImageNet, our method may he combined with those works aiming at state-of-the-ar performance through the potential on-chip fine-tuning after deployment.
E.2 DISCUSSION OF THE METHOD
The proposed OPZO is built online training methods to deal with the spatio-temporal locality problem of BP(TT) for neuromorphic computing and pave the path to on-chip SNN training. While the pseudo-zeroth-ordee formulation can be applied non-online scenarios, e-g-, BPTT, we do not focus on it because this is not our target (friendly for neuromorphic hardware and more biologically plausible) and it requires larger memory costs to maintain intermediate states through time for direct error propagation. Nevertheless, to further validate the feasibility of applying PZO to the setting without truncating the temporal gradient flow, we provide more results under this setting. Specifically, we leverage PZO to estimate gradients from the network output at time step t, to intermediate layers at time steps 1ú (l, S (4). To reduce the momentum costs, we share the momentum for the same interval between 4 and lj. i.e., there will be T feedback momentum representing the feedback matrix from o--outuut to (t, -ffcuuues (k = 0, ..... . - 1). We perform noise injection to all time steps and update the momentum feedback matrices based on noises and network outputs. Without the requirement for memory-efficien efficient online training, we can also adopt BN (along all time steps) for networks. As shown in Table 13 PZO can be effectively applied this setting.
We consider node perturbation instead of weight perturbation because it improves the latter with smaller variance (Lillicrap al. (2020) and is more biologically plausible with a better analog to three-factor Hebbian learning (Frémaux & Gerstner 2016) as discussed in Section 4.3. This is friendly for neuromorphic hardware that supports three-factor rules (Davies 202T). While weight perturbation may be conceptually easier, it is less effective to optimize neural networks and has hardly been adopted in zeroth-ordes methods for neural networks (Jiang al., 2024) except in specially designed fine-tuning settings (Malladi et al. 2023)
While this paper mainly considers SNNs, the proposed pseudo-zeroth-orde formulation can also be applied to ANNs. If we consider the general computer. there can be techniques to reduce the memory overhead of the momentum feedback, such as low-rank approximation. only saving some output vectors and recomputing the matrix by re-drawing the perturbation with random seeds, etc. This paper mainly focuses on neuromorphic computing, and we leave the extension to more settings as future work.
E.3 RELATION NEUROSCIENTIFC EVIDENCE
In this section, we discuss more on the neuroscientific evidence for noise injection and top-down signals in our method.
Noise injection. The biological systems are inherently noisy and has long been recognized that noise can be utilized as resource for computation and learning (Seung. 2003 Fiete & Seung. 2006) Maass, 2014 Lillicrap et al. (020). Considering the perturbation with noise injection, it can be related to stochastic synaptic transmission (Seung) 2003) or "empiric" synapses carrying perturbing input from another part of the brain (Fiete & Seung 2006). Such mechanisms provide the biological basis for perturbation learning, which is believed to be employed by the brain for some kinds of learning (Lillicrap et al. 2020). Our work builds on this zeroth-order perturbation, while introducing momentum feedback solve the large variance problem of it.
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Top-down feedback. In the three-factor Hebbian learning, synaptic updates modulated by reward-prediction errors (RPE) and can be gated by feedback (FB) from higher brain regions through top-down feedback connections, leading to the update rule Aw;,j B f((a.) fa(,)) RPE FB, (Roelfsema & Holtmaat] 2018). Anatomically, feedback projections originate from higher cortical areas and mostly provide input superficial (1.1-1.3 and deep (1.5) layers of lower sensory areas, targeting apical dendrites of pyramidal neurons and specific microcircuits (Roelfsema & Holtmaat 2018). These feedback pathways are thought to play a key role in gating plasticity, credit assignment, and context-dependent modulation. The momentum feedback connections in our method are analogous to these top-down feedback pathways that modulate prediction errors, providing basis for rules similar to three-factor Hebbian learning. Notably, the update of our feedback connections depends only on local pre- and post-synaptic, maintaining the simplicity and biological plausibility of the learning process.
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Reviews:
[{"review_id": "lFBK3NsuCE", "reviewer": "Reviewer_vgGd", "summary": "This paper introduces Online Pseudo-Zeroth-Order (OPZO) training, a novel method for supervised learning of Spiking Neural Networks (SNNs). OPZO aims to be more biologically plausible and hardware-friendly than standard backpropagation (BP). It achieves spatial credit assignment using only a single forward pass with injected noise and direct top-down feedback signals via momentum-based feedback connections, avoiding the biologically implausible weight symmetry and separate forward-backward phases of BP. Experiments on various datasets show OPZO achieves competitive accuracy compared to spatial BP while being estimated to have lower computational costs on neuromorphic hardware.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  The proposed momentum feedback connections successfully address the high variance problem typical of zeroth-order methods, enabling effective training where a basic single-point zeroth-order method fails. This is supported by theoretical analysis and empirical results showing gradient variances comparable to BP.\n\n2.  The method is designed with neuromorphic computing constraints in mind. It requires only one forward pass, enables parallel updates, and aligns with three-factor Hebbian learning rules, potentially leading to lower memory and operation costs for on-chip training compared to BP.", "weaknesses": "1.  The biological plausibility claims lack specificity. The paper frequently states that OPZO is more biologically plausible but does not sufficiently elaborate on how specific components, like the momentum feedback connections, map to known biological mechanisms. A more detailed discussion comparing the algorithm's components (e.g., noise injection, top-down signals) to neuroscientific evidence would strengthen this claim.\n\n2.  Theoretical grounding for momentum feedback could be stronger. While Proposition 4.1 analyzes variance reduction, the paper acknowledges that the momentum-based Jacobian estimation can introduce bias. Proposition 4.3 discusses conditions for providing a descent direction, but it would be strengthened by empirical analysis of this bias during training on complex tasks to show it does not hinder convergence.\n\n3.  Integration with local learning is loosely defined. The method of combining OPZO with Local Learning (LL) and Intermediate Global Learning (IGL) is described at a high level. The paper states that for LL, \"we assume the weight symmetry for propagating errors,\" but provides limited justification for this assumption in a biologically plausible or hardware-friendly context. A more detailed mechanism for how these global and local signals are integrated without violating the intended constraints would improve the methodology.\n\n4. The experiment with a 9-layer convolutional network uses Local Learning and IGL. However, the results in Appendix D.5 show that pure OPZO without residual connections performs poorly on deeper networks. This suggests the core method may have scalability limitations, which are mitigated by auxiliary techniques. A clearer discussion of this dependency and more experiments probing the depth limitations of the core OPZO method would provide a more complete picture.\n\n5.  Hardware Cost Analysis is Speculative. The analysis of computational costs (Table 5) is a theoretical estimate for potential neuromorphic hardware. The paper states that \"no implementation on neuromorphic hardware is included due to our limited access,\" which is acknowledged as a limitation. Concrete measurements or more detailed simulations based on specific hardware architectures (e.g., Loihi, SpiNNaker) would make the claims about efficiency much stronger.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SMXw447E1W", "reviewer": "Reviewer_bmzH", "summary": "This work proposes a pseudo-online zeroth-order training method that requires only a single forward pass, unlike existing approaches that rely on two forward passes. The authors achieve this by decoupling the loss function from the model and using the error signal (potentially a vector) computed at the final layer to update the model parameters. Additionally, a momentum feedback mechanism is introduced, which utilizes feedback signals from previous iterations to refine parameter updates. The authors claim that these modifications lead to improved performance and reduced variance in the training signal.\n\nExperimentally, the proposed method demonstrates enhanced performance across multiple datasets and is compared against a range of baseline methods.\n\nI am not an expert in zeroth-order optimization methods for SNN models, although I do have experience working with spiking neural networks. Therefore, I am assigning my score with low confidence.", "questions": "Please see the weaknesses section.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The presentation of the paper is clear and well-organized.\n\nThis work addresses an important issue: the high variance problem in zeroth-order optimization methods.\n\nThe proposed decoupling approach is particularly interesting and provides a novel perspective on improving training stability and efficiency.", "weaknesses": "My main concern is that existing backpropagation or surrogate-gradient-based methods already achieve strong performance in similar settings. It is therefore not entirely clear why one should prefer the proposed zeroth-order approach over these established alternatives.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FsFxZXRw4h", "reviewer": "Reviewer_EGrJ", "summary": "This paper proposes online pseudo-zeroth-order (OPZO) training for spiking neural networks. Existing zero-order optimization methods have large variances. To address this issue, the authors introduced a pseudo-zero-order formulation and used momentum feedback connections. Moreover, to apply this method to SNNs, the authors replace the gradient estimation in the OTTT method with the proposed method. To demonstrate the effectiveness of their method, the authors conducted experiments on static and neuromorphic datasets.", "questions": "See Weakness.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The experimental results demonstrate that the proposed method effectively reduces variance.\n2. Experimental results demonstrate that the proposed method exhibits robustness against different gradient noise injections.", "weaknesses": "1. The novelty of introducing a momentum mechanism in error propagation is limited.\n2. It does not seem that the proposed method takes into account the unique properties of SNNs.\n3. The performance of the proposed method is not consistently superior. The proposed method performs worse than OTTT [1], especially on static datasets such as CIFAR and ImageNet. The authors did not even compare the performance of the proposed method against OTTT on ImageNet. Furthermore, there is a lack of comparisons to other SOTA online learning or acceleration methods for SNNs.\n\n\n```\n[1] Online Training Through Time for Spiking Neural Networks. NeurIPS. 2022.\n```", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yBmopahKM4", "reviewer": "Reviewer_Pjav", "summary": "The paper proposes a method named OPZO for training Spiking Neural Networks (SNNs), which replaces spatial backpropagation with a single noised forward pass and momentum-updated feedback connections, aiming to achieve efficient and hardware-friendly credit assignment.", "questions": "1.OPZO and all compared methods are built upon the OTTT framework, which truncates the gradient flow across the time dimension. Is this a fundamental limitation for OPZO? Could OPZO be applied in a setting without truncating the temporal gradient flow (e.g., with full STBP)?\n\n2.The paper mentions that the pseudo-zeroth-order method introduces bias but does not sufficiently analyze its impact on the final performance. Could you provide more theoretical analysis or experimental evidence on how this bias affects training stability and final accuracy?\n\n3.Could you explain the rationale for removing the $1/\\alpha$ factor in the practical implementation and discuss its impact on convergence?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The OPZO method is novel and effective, skillfully addressing the high variance problem of traditional zeroth-order methods by retaining the first-order information of the loss function. Through extensive experiments, the paper demonstrates that OPZO's performance is significantly superior to other bio-plausible methods (like DFA) and can approach or even match the level of backpropagation (BP), offering a highly promising direction for efficient on-chip SNN training.", "weaknesses": "1.The provided ImageNet experiment is limited to fine-tuning a pre-trained network, which fails to demonstrate its performance on large-scale networks and datasets trained from scratch. For large-scale networks, OPZO might heavily rely on Local Learning to achieve performance comparable to BP.\n\n2.Table 8 shows that OPZO requires an extremely large $\\lambda$ for stable training. According to Equation 6, this implies that the feedback matrix M is updated very slowly. This raises the question: what would the performance be if a static M (i.e.,$\\lambda=1$ after an initial phase) were used?\n\n3.The paper lacks details on the initialization of the feedback matrix M, which weakens the method's reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)