# spiking_transformer_with_experts_mixture__423c0597.pdf

Spiking Transformer with Experts Mixture

Zhaokun Zhou1,2 , Yijie Lu1 , Yanhao Jia3,7, Kaiwei Che1,2, Jun Niu1, Liwei Huang4,2, Xinyu Shi5,4, Yuesheng Zhu1, Guoqi Li6,2, Zhaofei Yu5,4 , Li Yuan1,2

1School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University 2Peng Cheng Laboratory 3 College of Computing and Data Science, Nanyang Technological University 4School of Computer Science, Peking University 5Institute for Artificial Intelligence, Peking University 6Institute of Automation, Chinese Academy of Sciences 7Deep Neuro Cognition Lab, I2R and CFAR, Agency for Science, Technology and Research

Spiking Neural Networks (SNNs) provide a sparse spike-driven mechanism which is believed to be critical for energy-efficient deep learning. Mixture-of-Experts (Mo E), on the other side, aligns with the brain mechanism of distributed and sparse processing, resulting in an efficient way of enhancing model capacity and conditional computation. In this work, we consider how to incorporate SNNs spike-driven and Mo E s conditional computation into a unified framework. However, Mo E uses softmax to get the dense conditional weights for each expert and Top K to hard-sparsify the network, which does not fit the properties of SNNs. To address this issue, we reformulate Mo E in SNNs and introduce the Spiking Experts Mixture Mechanism (SEMM) from the perspective of sparse spiking activation. Both the experts and the router output spiking sequences, and their element-wise operation makes SEMM computation spike-driven and dynamic sparse-conditional. By developing SEMM into Spiking Transformer, the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP) are proposed, which performs routing allocation for head-wise and channel-wise spiking experts, respectively. Experiments show that SEMM realizes sparse conditional computation and obtains a stable improvement on neuromorphic and static datasets with approximate computational overhead based on the Spiking Transformer baselines.

1 Introduction

The spiking neural networks (SNNs) are regarded as the third generation of neural networks [1], distinguished by biological plausibility [2], spike-driven characteristic, and low power consumption. SNNs emulate the dynamics of biological neurons at a microscopic level, utilizing asynchronous binary spikes for information transmission. The membrane potential of spiking neurons in SNNs is only updated upon the arrival of spikes, avoiding calculations of zero values. The inherent features make SNNs promising candidates for low-energy consumption on neuromorphic hardware, such as True North [3] and Loihi [4]. There are lots of architectures in SNNs include Spiking Recurrent Neural Networks [5], Res Net-like SNNs [6 9], Spiking Graph Neural Networks [10], and Spiking Transformers [11 13]. Spiking Transformers stands at the forefront. Spikformer [11] introduces Spiking Self-Attention (SSA). The Spike-Driven Transformer [12] introduces Spike-driven Self-

Equal Corresponding author

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Attention. Other works explore the Spiking Transformer in terms of structural improvements [14 17], training methods [18], and different tasks [19], respectively.

Mixture-of-Experts (Mo E) [20, 21] is known for allowing each expert to learn specific tasks or features, showing better performance, conditional computing and dynamic adaptability, which are crucial features in the brain mechanism [22, 23]. In this work, we are committed to exploring the effective integration of Mo E and Spiking Transformer. As shown in Fig.1(a), Mo E introduces a large number of parameters based on the original Transformer, and the conditional computation is achieved by calculating the routing probability of each token on each expert through the softmax function. Selecting Top-K experts based on the routing probability, Mo E achieves hard sparsification. However, SNN calculations need to avoid multiplication and cannot use complex softmax functions. The features in SNNs are dynamically sparse and do not require additional Top K. All the parameters of the expert must be loaded, which aggravates the difficulty of neuromorphic chip deployment. These factors make porting Mo E to SNNs non-trivial. To tackle this problem, we develop the Spiking Experts Mixture Mechanism (SEMM), as shown in Fig.1(b), a universal SNN-Mo E paradigm with the following three main features. 1) The SEMM is spike-driven. The outputs of the expert and the router are spike sequences, and the element-wise operation between them conforms to the SNN characteristics, i.e., avoiding multiplication. 2) SEMM leverages the sparse spiking activation of SNNs to achieve dynamic conditional computation of Mo Es, which is more flexible than the fixed hard sparsification of Artificial Neural Network Mo E (ANN-Mo E). 3) With reasonable parameter count settings, SEMM enables Spiking Transformers to achieve stable performance gains with negligible overhead.

Element-wise operation

Expert 1 Expert 2 Expert 3 Expert 4

P=0.45 P=0.36

Element-wise Add

Spiking Expert 1

Spiking Expert 2

Spiking Expert 3

Spiking Expert 4

Spiking Input

(a) ANN-Mo E (b) SEMM

Figure 1: ANN-Mo E and Spiking Experts Mixture Mechanism (SEMM). N denotes the length of image patches.

Based on SEMM, we modify the Spiking Self-attention (SSA) and Multi-layer Perceptron (MLP) of Spiking Transformers to obtain the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP). EMSA treats each head of SSA as an expert, computes respective attention, and employs a temporal-aware router to integrate attention. As for Multi-layer Perceptron (MLP), it is common to substitute the entire MLP with Mo E [24], which leads to better performance but a significant increase in the overall parameter number. EMSP implements a channel-wise Mo E within MLP to overcome the shortcoming. EMSA and EMSP can be inserted directly and seamlessly into existing Spiking Transformer variants [11, 14, 12]. Our work contributes in three main aspects:

We introduce the Spiking Experts Mixture Mechanism (SEMM), a universal SNN-Mo E paradigm. SEMM is spike-driven, capable of efficient dynamic sparse conditional computation.

Based on SEMM, we develop the Experts Mixture Spiking Attention (EMSA), whose information from all head-wise experts is selectively integrated through a temporal-aware router. We restructure MLP by a channel-level spiking-sparse SEMM, named the Experts Mixture Spiking Perceptron (EMSP). They can seamlessly replace self-attention and MLP in Spiking Transformers.

Extensive experiments demonstrate the stable performance improvement of SEMM on both static and neuromorphic datasets. Notably, Spike-driven Transformer-8-512 with SEMM achieves a remarkable 76.62% accuracy on Image Net with 4 time steps, surpassing the baseline (74.57%).

2 Related Work

Deep Spiking Neural Networks and Spiking Transformers. Spatio-temporal backpropagation(STBP) [25] directly trains SNNs by performing backpropagation on both spatial and temporal domains. Temporal backpropagation [26] computes the gradients of the timings of existing spikes for the membrane potential at the spike timing. Treshold-dependent batch normalization (td BN) [8] is used to extend the network depth. SEW-Res Net [7] proposed the spiking element-wise residual for SNNs. Spikformer [11] firstly converts all the components of Vision Transformer (Vi T) into

(a) Spiking Transformer (c) EMSP

DWC Depth-wise Conv

Element-wise Add Hadamard Product

Figure 2: (a) The overview of Spiking Transformer. (b) The Experts Mixture Spiking Attention (EMSA). (c) The Experts Mixture Spiking Perceptron (EMSP). EMSA and EMSP can directly replace SSA and MLP in (a).

spike-form and pioneers the field of SNN-Transformer. Spike-driven Transformer [12] goes further by introducing linear spike-driven self-attention. Spikingformer [14] proposes a hardware-friendly spike-driven residual learning architecture. Besides, Masked Spiking Transformer [27] combines SNNs and Transformers from the perspective of the ANN-to-SNN conversion. However, the SNN Transformer architecture that can bring SNNs superiority into full play is still ongoing research.

Mixture-of-Experts. The Mixture-of-Experts (Mo E) [28, 29] combines the predictions of multiple specialized experts, which is effective in handling high-dimensional data and complex problems. Researchers explore diverse gating mechanisms [30, 31], optimizing expert allocation strategies [32, 33], and enhancing the scalability of Mo E models [34, 35]. Sparsely-Gated Mixture-of-Experts [30] adopted Mo E into architectures such as the Long Short-Term Memory (LSTM) [36], showcasing effectiveness in Language Modeling. The Transformer also benefits from Mo E with the substitution of the Multi-layer Perceptron (MLP) [24, 37]. Switch Transformer [35] has scaled the models with trillions of parameters. Currently, there is no existing work on Mo E with Spiking Transformers.

3 Methodology

3.1 Preliminaries and Overall Architecure

Spiking neuron is the basic unit of SNNs. For the dynamics of the Leaky Integrate-and-Fire (LIF) neuron used in this work, the t th-time-step membrane potential U[t] is equal to the sum of the state potential H[t 1] at the previous time step and the input X[t]. When membrane potential exceeds the threshold uth, the neuron will fire a spike, otherwise, it remains inactive. Consequently, the output S[t] only contains binary values, either 1 or 0. Hea( ) is a Heaviside function that satisfies Hea (x) = 1 when x 0, otherwise Hea (x) = 0. H[t] is the temporal state output, and Vreset denotes the reset potential after a spiking event. β < 1 determines the rate of decay. If the neuron remains inactive, the potential U[t] decays towards H[t] over time. LIF can be described as: U[t] = H[t 1] + X[t], (1) S[t] = Hea (U[t] uth) , (2) H[t] = Vreset S[t] + (βU[t]) (1 S[t]) . (3)

Spiking Transformer[11, 14, 12] baselines contain Patch Splitting (PS), Relative Position Embedding (RPE), Spiking Self Attention (SSA) (e.g., Spiking Self-Attention [11] and Spike-driven Self-Attention[12]), MLP and linear classification head, as shown in Fig. 2(a). Given a 2D image sequence I RT C H W , where T, C, H, and W denote time-step, channel, height and width, the PS splits it into a sequence of N flattened spike patches x with D dimensional channel. A Convolution-Batch Norm-LIF block generates Relative Position Embedding (RPE):

X0 = PS (I) + RPE, (4)

X l = SSA(Xl 1) + Xl 1, (5)

Xl = MLP(X l) + X l, (6) Y = CH(GAP(XL)), (7)

where the X0 is fed into the L-blocks and each block consists of a SSA and a MLP. X l, Xl are spike sequences and l = 1...L is layer. A Global Average-pooling (GAP) is utilized on the XL and the linear Classification Head (CH) to output the prediction Y. See Appendix. A for more details.

3.2 Spiking Experts Mixture Mechanism

Before exploring the adaptation of spiking Mo E in SNNs, we first review Mo E in ANNs. An ANNMo E layer typically comprises a set of m experts EA = {E1, E2, ..., Em}, along with a router RA for selecting the corresponding experts. Given an input sequence XA, the resulting output can be expressed as the sum of the Top-K selected experts from m candidates using a router:

k=1 Rk(XA) Ek(XA), (8)

R(x) = Top K(softmax(XAWR, K)), (9)

Top K(v, K) = v if v is in the top K elements 0 otherwise (10)

where WR is the router weight matrix. The Top K( , K) together with the softmax( ) sets all elements of the routing vector to zero except the largest top K values. The sparse conditional computing is obtained from the selecting of Top K and the different routing weights of the softmax, while K is usually taken as 0.5 m. We argue that the ANN-Mo E is not suitable for SNN for two main reasons. First, the float-point routing-expert and the softmax which involve exponentiation and division, do not adhere to the computation principles of SNNs. Second, SNN experts are inherently highly sparse, so the hard sparsification approach of additional Top K is unnecessary. An eventtriggered spike-based sparse conditional computation on asynchronous neuromorphic chips is needed more than Top K. To bridge these gaps, we introduce a generalized representation of the Spiking Experts Mixture Mechanism (SEMM), which is as follows, SEMM(E, R, F( )) = F(E, R), (11) where E = {E1, E2, ..., Em} represents the spiking sequence of m spiking experts in the Spiking Transformer, and R {0, 1}T N m represents the spiking router for allocating computation. F( ) denotes the element-wise form of the operation between the router and the expert output spiking sequence, i.e., Hadamard product and addition.

As a Mo E mechanism specifically designed for SNNs, SEMM has the following three significant advantages. i) Spike-driven. The SEMM is spike-driven, which is important for SNNs. Due to the spike-driven computation mode of experts, i.e., Spiking Self-attention and Spiking-MLP, SEMM computations are triggered by sparse spikes of experts and require only synaptic manipulation. For example, the Hadamard product between the spiking signals R and E is equivalent to mask. ii) Sparse-spiking conditional computation. The SEMM subtly utilizes the sparse activation of the spiking routers for the conditional computation of Mo E. SEMM has a variable sparsity when dealing with different data. Additionally, SEMM does not suffer from the load imbalance problem in ANNMo E, i.e., Top K selects fixed number of experts. The sparse conditional computation is distributed to each expert. iii) Efficient computation. Unlike loading with multiple heavy expert modules of ANN-Mo E, the SEMM has comparable parameters and operations to the previous SSA and MLP of Spiking Transformers. These are further discussed in Sec. 3.5. Without loss of generality, we use the mainstream architecture of Spiking Transformer for SEMM embedding in Sec. 3.3 and Sec. 3.4.

3.3 Experts Mixture Spiking Attention

We begin by reviewing the processing of Spiking Self-Attention (SSA). Different from vanilla ANNTransformers [38], it discards the softmax normalization for the attention map. The SSA mechanism

can be described by the following equation:

Q = SN Q(L-BNQ(X)), K = SN K(L-BNK(X)), V = SN V(L-BNV(X)), (12)

A = SSA(Q, K, V) = SN(QKTV s), for Spiking Self-Attention SN (SUMc (Q K)) V, for Spike-Driven Self-Attention (13)

where SN represents the spiking neuron and L-BN represents that the features pass sequentially through Linear and Batch Norm. s is the scaling factor and Q, K, V {0, 1}T N D are in spikeform. A {0, 1}T N D is the spiking output of SSA. is the Hadamard product, and SUMc( ) denotes the sum of each column. As shown in Fig 2(b), the Experts Mixture Spiking Attention (EMSA) based on SEMM and SSA is formulated as:

Am = SSA(Qm, K, V), (14) E = {A1, A2, ..., Am}, (15) R = SN(BN(XWR)) = {r1, r2, ..., rm}, (16)

SEMM(E, R, F( )) =

i=1 ri Ai, (17)

EMSA = SN(BN(SEMM(E, R, F( )))Wo), (18)

where Am {0, 1}T N d is the output of each SSA experts, and WR RD m is the router weight matrix. R {0, 1}T N m is the routing sequence. We set d D by default to avoid introducing too many parameters. The operation F( ) between expert-router is computed by masking the expert using router pair-by-pair and then summing them up. The output of EMSA is obtained by performing matrix multiplication between SEMM(E, R, F( )) and synaptic weight Wo Rd D, the same as the last layer of SSA block in Spiking Transformers.

3.4 Experts Mixture Spiking Perceptron

As illustrate in Fig. reffig:method, the Experts Mixture Spiking Perceptron (EMSP) launches the channel-wise SEMM within MLP. The architecture can be written as follows:

E = SN(DWC(SN(BN(XW1)))) {0, 1}T N m (19)

R = SN(BN(XWR)) {0, 1}T N m, (20) SEMM(E, R, F( )) = E R, (21)

EMSP = SN(BN((SEMM(E, R, F( ))Wo)) {0, 1}T N D, (22)

where W1, WR RD m is the weight matrix of first layer in MLP and router, respectively. We integrate a 3 3 Depth Wise Convolution layer DWC to capture local features on each channel expert, which has fewer parameters and is computationally efficient compared to original convolution. Wo Rm D is the wight matrix of output layer. The original Spiking Transformer s MLP would have a D to 4 D channel dimension increase after the first layer and the second layer would reduce the dimension back to D. In EMSP we set m to (8//3) D to try to match the parameter number of the original MLP. EMSP integrates sparse routing of multiple channel-wise experts. It can also be viewed as the channel-wise gating, which is suitable for temporal information processing, allowing information to flow unimpeded through potentially many time steps [39]. The gate (router) branch and expert branch can be also regarded as incorporating the Spike Element-Wise (SEW) [7] residual block into EMSP. Channel-wise convolution and element-wise (Hadamard) products only introduce a minor increase in computational cost. By appropriately configuring the number and dimensions of expert networks, our EMSP achieves more efficient computation compared to the original spiking MLP. More details of the computational overhead are given in the next sub-section.

3.5 Characteristics of SEMM

We explain each of the three advantages of SEMM, i.e., Spike-driven, Sparse-spiking conditional computation and efficient computation.

Spike-driven has the following formal definition, meaning that the gathering of input current is initiated by sparse spikes released from presynaptic neurons:

Definition 1. In SNN, the operation is spike-driven if the input currents satisfy the following form,

j wi,jsj[t] = X

j,sj[t] =0 wi,j, (23)

where Ii[t] is the input current of the i-th postsynaptic neuron at time step t, sj[t] {0, 1} is the spike output of the j-th pre-synaptic neuron, wi,j is the weight of the synaptic connection from j to i.

For EMSA, the input current for the LIF in Eq. 18 at a specific time step t is given by:

I[t] = SEMM(E[t], R[t], F( ))Wo =

i=1 ri[t] Ai[t]Wo, (24)

i=1 ri[t] X

l ai,p,l[t]wl,q = X

i,l (ri[t] ai,p,l[t]) =0

where I[t] has a dimension of P Q, with p and q represent the p-th row and q-th column, respectively. EMSA is essentially spike element-wise addition after masking operation on SSA, consistent with the characteristics of spike-driven. For EMSP, the operation between two spiking sequences E and R corresponds to the logical AND function [7], which also conforms to the spike driven,

I[t] = SEMM(E[t], R[t], F( ))Wo = (E[t] R)[t]Wo, (26)

Ip,q[t] = X

l ep,l[t]rp,l[t]wl,q = X

l (ep,l[t] rp,l[t]) =0

10 15 20 Params (M)

Accuracy (%)

3 4 5 Params (M)

CIFAR10-DVS

Spikformer Spike-driven Transformer Spikingformer

Figure 3: Comparison of parametersaccuracy for different Mo Es. Methods of each line from left to right correspond to: 1. the baseline, 2. EMSP, 3. ANN-router with four heavy MLP experts, 4. Spiking-router with four heavy MLP experts, respectively. The last two use softmax and Top K (K = 2).

Sparse-spiking conditional computation means using spiking routers to dynamically allocate the computation in temporal and spatial dimension. More analysis is detailed in experiments.

Efficient computations means that SEMM approximates SSA and MLP in terms of number of parameters and theoretical synaptic operations. Tab. 1 demonstrates the computation load for EMSA, EMSP versus SSA and MLP. In terms of parameter number, despite having a routing layer m D, EMSA has a smaller number of parameters than SSA when the number of experts is within a reasonable range ( 2). EMSP additionally introduces a 3 3 depthwise convolution, and the number is slighter higher than MLP. Due to the similarity of the actual spiking rates, EMSA is smaller than SSA on the TND2 term, while the calculation of the additional introduced by SEMM is on the TND term (much smaller than TND2) and therefore can be ignored. The situation is similar on EMSP, with depth-wise convolution adding a slight computational overhead. The design of EMSP is different from ANN-Mo E, which selects multiple heavy MLPs as experts. We demonstrate it s validity in Fig. 3, i.e., it performs better while the number of parameters is much smaller than ANN-Mo E. 4 Experiments

4.1 Sparse Conditional Computation Analysis

We analyze the average spiking rate (ASR) of routers for EMSA and EMSP on the Image Net validation set, which is shown in Tab. 2. The SD-Transformer-8-512 is used for the analysis. The ASR of EMSA is around 0.5, which is comparable to the regular Top K setting of ANN-Mo E. The ASR of EMSP is low and gradually decreases as the block deepens, compared to the fixed Top K hard sparse in ANN-Mo E, SEMM fully reflects the advantage of SNN dynamic sparse conditional computation. To further verify the spiking router, we ablate it, i.e., cancel it in EMSA and EMSP, as

Table 1: Number of parameters and theoretical synaptic operations. R, b R, R and R denote the average spike firing rates (the proportion of non-zero elements in the spike matrix) in various spike matrices. T, N, and D are the time step, sequence length, and channel dimension of the input features, respectively. d is the channel dimension of Am and m is the number of experts. The details are provided in the Appendix. B.

SSA 4D2 4RTND2 + 2 b RTN 2D

EMSA (1 + 1/m)D2 + (2d + m)D (1 + 1/m)RTND2 + (2d + m) RTND + (D + md)RTN 2

MLP 8D2 8RTND2

EMSP 8D2 + 24D 8RTND2 + 24 RTND

Spikformer Spike-driven Transformer Spikingformer 95.0

Accuracy (%)

Baseline EMSA w/o Router EMSA EMSP w/o Router EMSP

Spikformer Spike-driven Transformer Spikingformer 76.8

Accuracy (%)

Baseline EMSA w/o Router EMSA EMSP w/o Router EMSP

Figure 4: Ablation study on the EMSA and EMSP router.

shown in Fig. 4, the accuracy decreases significantly after canceling the router, and it is even lower than the baseline model on the Spikingformer. Directly analyzing the conditional computation of SEMM is challenging, we therefore use visualizations to illustrate this intuitively. As show in Fig. 5, for the same image, each expert s router assigns a different computational region, e.g., the third router filters the background of the expert s features, while the second router assigns the computation to the foreground target. The computation allocation of the spiking router to the irregular object "snake" can be seen that the router is highly dynamic and effective. See Appendix. D for more samples. In addition, as shown in Fig. 6, we also report the ASR of spatial-temporal locations of routers in different kinds of images. As can be seen by the difference in average firing rates, spiking router has a dynamic adjustment of ASR processing different kinds of images, further illustrating its data-dependent conditional computation property.

Table 2: The average spike rate of EMSA and EMSP router in 8 blocks testing on the Image Net.

Block0 Block1 Block2 Block3 Block4 Block5 Block6 Block7

EMSA 0.64 0.68 0.49 0.52 0.47 0.43 0.51 0.50

EMSP 0.25 0.30 0.33 0.27 0.20 0.10 0.05 0.02

4.2 Results on various Datasets

baseline 2 4 6 8 12 Expert Num

Accuracy (%)

baseline 2 4 6 8 12 Expert Num

Spikformer Spike-driven Transformer Spikingformer

Figure 8: Accuracy with different experts number.

We conduct experiments on static datasets, i.e., Image Net [40] and CIFAR [41], and neuromorphic datasets, i.e., CIFAR10-DVS [42], DVS128 Gesture [43] to verify the effectiveness of SEMM. See Appendix. C for more details. Image Net results are shown in Tab. 3 which mainly compares SEMM with the Spiking Transformer baselines. At slightly lower model parameter counts, SEMM is steadily superior to baselines. For instance, Spikformer-8-512 with SEMM is 2.55% higher than the baseline with 28.22M parameters, Spike-driven Transformer-8-384 with SEMM obtain 2.05% improvement. SEMM on Spikingformer presents similar findings.

Figure 5: Visualization of routers as masks. The mask position (black) indicates router of 0 here and the background image is the same for each subplot. We show the dynamic sparsity of spiking router for different experts (horizontal direction) and time steps (vertical direction).

Figure 6: Average spiking rate of different kinds of images in the Image Net validation set in the spatial-temporal dimension. The height of the cube is the time step. (a) Japanese spaniel. (b) Volcano.

Spikformer Spike-driven Transformer Spikingformer 95.0

Accuracy (%)

Baseline +EMSA +EMSP +SEMM

Spikformer Spike-driven Transformer Spikingformer 76.8

Accuracy (%)

Baseline +EMSA +EMSP +SEMM

Figure 7: Ablation study of EMSA and EMSP module.

The experimental results on CIFAR, CIFAR10-DVS, and DVS128 Gesture are shown in Tab. 4. These four datasets are relatively small. We basically keep the experimental setup in [11, 12, 14], including the network structure, training settings, etc., and details are given in the Appendix. C.1. SEMM has demonstrated stable performance improvements across various datasets when integrated

Table 3: Results on Image Net-1k. Model-L-D represents a model with L encoder blocks and D channels.

Methods Architecture Param (M) Time Step Top-1 Acc (%)

SEW Res Net[7] SEW-Res Net-34 21.79 4 67.04 SEW-Res Net-101 44.55 4 68.76 SEW-Res Net-152 60.19 4 69.26

MS-Res Net[9] MS-Res Net-18 11.69 6 63.10 MS-Res Net-34 21.80 6 69.42 MS-Res Net-104 77.28 5 76.02

Spikformer[11] Spikformer-8-384 16.81 4 70.24 Spikformer-8-512 29.68 4 73.38

Spikformer + SEMM Spikformer-8-384 16.05 4 72.86 Spikformer-8-512 28.22 4 75.93

Spike-driven Transformer[12] SD-Transformer-8-384 16.81 4 72.28 SD-Transformer-8-512 29.68 4 74.57

SD-Transformer + SEMM SD-Transformer-8-384 16.05 4 73.93 SD-Transformer-8-512 28.22 4 76.62

Spikingformer[14] Spikingformer-8-384 16.81 4 72.45 Spikingformer-8-512 29.68 4 74.79

Spikingformer + SEMM Spikingformer-8-384 16.05 4 73.58 Spikingformer-8-512 28.22 4 76.03

Table 4: Results on CIFAR10-DVS, DVS128 Gesture, and CIFAR.

Methods CIFAR10-DVS DVS128 Gesture CIFAR-10 CIFAR-100

T Acc T Acc T Acc T Acc

td BN [8] 10 67.80 40 96.90 6 93.20 - - PLIF [44] 20 74.80 20 97.60 8 93.50 - - DIET-SNN [45] - - - - 5 92.70 5 69.70 Dspike [46] 10 75.40 - - 6 94.30 6 74.20 DSR [47] 10 77.30 - - 20 95.40 20 78.50

Spikformer [11] 10 78.90 10 96.90 4 95.19 4 77.86 16 80.90 16 98.30

Spikformer + SEMM 10 82.32 10 97.56 4 95.78 4 79.04 16 82.90 16 98.63

Spike-Driven Transformer [12] 10 78.90 10 96.90 4 95.60 4 78.40 16 80.00 16 99.30

Spike-Driven Transformer + SEMM 10 81.10 10 97.56 4 96.12 4 80.26 16 82.42 16 99.30

Spikingformer [14] 10 79.90 10 96.20 4 95.81 4 79.21 16 81.30 16 98.30

Spikingformer + SEMM 10 80.70 10 96.88 4 96.16 4 80.24 16 82.10 16 98.56

into different Spiking Transformer baselines. Specifically, SEMM achieves SOTA on CIFAR-10 (96.16%), CIFAR-100 (80.26%), CIFAR10-DVS (82.9%) and DVS128 Gesture (99.3%).

4.3 Ablation Study and Hyperparameter Sensitivity

Module ablation. EMSA and EMSP together improve the performance of the baseline, as shown in Fig. 7. The use of both EMSA and EMSP alone is better than baseline, illustrating their respective superiority.

Experts number. We examine the utility brought by different numbers of experts of EMSA. The results of Spiking Transformer baselines on CIFAR are presented in Fig. 8. It indicates that within a certain range of expert numbers, the results can still be competitive and robust. Among these, we select 4 experts as the parameter setting for our final application.

5 Conclusion

In this work, we explored the feasibility of adapting Mo E in Spiking Neural Networks and formulated an SNN-Mo E paradigm named the Spiking Experts Mixture Mechanism. Unlike the vanilla Mo E, which uses softmax and Top K hard sparse, SEMM implements dynamic conditional computation from the viewpoint of spiking sparse activation. With redesigned Router-Expert pairs and element-wise spike-driven operations, SEMM is computation-efficient and SNN-compatible. Embedded in Spiking Transformers, SEMM-based EMSA and EMSP can bring stable performance improvement on static and neuromorphic datasets. SEMM paradigm can inspire future exploration of high-performance, high-capacity Spiking Transformers. We hope SEMM can bring vitality to dynamic conditional computation and the design of next-generation architecture for SNNs. Future work will explore SEMM implementation in a wider range of tasks and larger SNN models.

6 Acknowledgments and Disclosure of Funding

We sincerely thank Dr. Wei Fang for his assistance with this work. This work is supported by grants from the National Natural Science Foundation of China (62088102, 62202014, 62332002, 62425101, 62236009, U22A20103, 62441606), Shenzhen Basic Research Program (No.JCYJ20220813151736001), the major key project of the Pengcheng Laboratory (PCL2021A13) and National Science Foundation for Distinguished Young Scholars (62325603). Computing support was provided by Pengcheng Cloud Brain.

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A Overall architecture of Spike-driven Transformer/Spikingformer with SEMM

We provide a detailed description of the overall structure of Spike-driven Transformer with SEMM, and Spikingformer with SEMM. Given a 2D image sequence I RT C H W , the Patch Splitting Module (PSM), consisted of three Convolution-Batch Normalization-SN-Maxpooling layers and one Convolution-Batch Normalization-Maxpooling layer, is used to projects and splits I into a sequence of N flattened spike patches s with D channels. The Relative Position Embedding (RPE) module, a Convolution-Batch Norm layer, is used to generate position embedding to get U0. SN( ) means the spike neuron layer. Then the U0 is passed to the L-blocks encoder. The encoder block consists of an Experts Mixture Spiking Attention (EMSA) and an Experts Mixture Spiking Perceptron (EMSP) block. Membrane-shortcut residual connections are applied in both the EMSA and EMSP blocks. A global average-pooling (GAP) is utilized on the processed feature from the encoder and outputs the D-dimension feature which will be sent to the fully-connected-layer classification head (CH) to output the prediction Y . The architecture can be written as follows:

u = PSM (I) , I RT C H W , x RT N D, (28)

s = SN(u), s RT N D (29)

RPE = BN(Conv2d(s)), RPE RT N D (30)

U0 = u + RPE, U0 RT N D (31)

S0 = SN(U0), S0 RT N D (32)

U l = EMSA(Sl 1) + Ul 1, U l RT N D, l = 1...L (33)

S l = SN(U l), S l RT N D, l = 1...L (34)

Sl = SN(EMSP(S l) + U l), Sl RT N D, l = 1...L (35) Y = CH(GAP(SL)), (36)

Following the Spiking Experts Mixture Mechanism and Spike-driven Self-Attention (SDSA) used in Spike-driven Transformer, the EMSA can be written as:

Q = SN Q(L-BNQ(X)), K = SN K(L-BNK(X)), V = SN V(L-BNV(X)), (37) A = SDSA(Q, K, V) = SN (SUMc (Q K)) V, (38)

The rest operate according to standard EMSA 18. The EMSP in Spike Driven Transformer is also much the same as the standard 22, except that no spiking neuron layer in the output layer.

B Computation Overhead Details of SEMM

The linear layers that generates m Query Qm have m( D2

m ) = D2 parameters. The linear layer for Key K has D2

m parameters. Generating Value V requires d D parameters. The output linear has d D parameters. The router linear layer transforms D to expert number m, and has m D parameters. The number of total parameters is (1 + 1/m)D2 + (2d + m)D. For operations, obtaining Query and Key needs (1 + 1/m)RTND2. Computing Value or the output of EMSA requires d RTND. The number of operations between Qm, K, V is (D + md)RRTN 2. The number of router matrix computation operations is m RTND. Due to the sparsity of SEMM and the smaller data dimensions, the number of element-wise operations for the router and expert is negligible.

The parameter number of W1, WR, Wo are all (8//3)D2. The number of parameters for 3 3 depthwise convolution is 3 3 (8//3)D = 24D. The all parameter number is 8D2 + 24D. The operation number of W1, WR, Wo are all (8//3)TND2. The number of operations for depthwise convolution is 24 RTND.

C Experiment Details

C.1 Experimental Setup

Our experimental framework closely aligns with the methodology outlined in [11]. All of Baseline s code is publicly available, and we abide by their credentials. For the Image Net-1K, we utilize a fixed timestep count of T = 4. The optimizer is the Adam W, with a batch size of 128 or 256 over the course of 310 training epochs. The learning rate is governed by a cosine-decay schedule, starting from an initial value of 0.0004. We incorporate a suite of standard data augmentation techniques, including random augmentation, mixup, and cutmix, into our training. For four small datasets, we adapt the SEMM to a variety of baseline models, following the precedents set by [11, 12]. For CIFAR, we maintain a timestep count of T = 4. For the neuromorphic datasets, we increase this to T = 10 and T = 16, respectively. Our experimental setup is consistent with each Spiking Transformer baselines, as detailed below.

Spikformer with SEMM. The training epoch is set to 310 for CIFAR, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 128 for CIFAR, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.001 for DVS128 Gesture and CIFAR10DVS. All of them are reduced with cosine decay. We follow [11] to apply data augmentation on DVS128 Gesture and CIFAR10-DVS. In addition, the network structures used in CIFAR-10, CIFAR10-DVS, and DVS128 Gesture are Spiking Transformer-4-384, Spiking Transformer-2-256 and Spiking Transformer-2-256.

Spike Driven Transformer with SEMM. The training epoch is set to 210 for CIFAR10/100 datasets, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 32 for CIFAR10/100, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.0003 for DVS128 Gesture and 0.01 for CIFAR10-DVS. The rest is consistent with Spikformer.

Spikingformer with SEMM The training epoch is set to 410 for CIFAR10/100 datasets, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 64 for CIFAR10/100, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.1 for DVS128 Gesture and CIFAR10-DVS. The rest is consistent with Spikformer.

C.2 Model Details

In our experiments, we use 8 NVIDIA-4090 GPUs for Image Net, and 1 NVIDIA-4090 GPU for other datasets. We adjust the value of membrane time constant τ in spike neuron when training models on DVS datasets. In direct training SNN models with surrogate function,

Sigmoid(x) = 1 1 + exp( αx) (39)

We select the Sigmoid function as the surrogate function with α = 4 in all experiments.

C.3 Additional Experiments

Table 5: Ablation study results on the time step.

Datasets Models Time-Step Top1-Acc(%)

CIFAR10/100

Spikformer + SEMM 1 94.11/74.67 2 94.87/78.49 4 95.78/79.04 6 95.99/79.29

Spike Driven Transformer + SEMM 1 94.97/76.77 2 95.58/78.49 4 96.12/80.26 6 96.48/80.87

Spikingformer + SEMM 1 94.54/77.16 2 95.42/78.85 4 96.16/80.24 6 96.67/81.22

The accuracy regarding different simulation time steps of the spiking neuron is shown in Tab. 5. At all the time steps, our Spiking Transformer with SEMM shows competitive performance. When the

time step is 1, our Spikformer with SEMM is 1.7% lower than the network with T = 4 on CIFAR10, and our Spikformer with SEMM with 1 time step still achieves 74.67% on CIFAR100. The above results show that Spikformer with SEMM is robust under fewer time steps.

In EMSP, the role of Deep Wise Convolution layer (DWC) is to independently extract features at the channel-level experts using a 3x3 receptive field, thereby enhancing representational capacity. The ablation study about DWC in Tab. 6 demonstrates its effectiveness.

Table 6: Ablation on Deep Wise Convolution layer. Model Module CIFAR100 CIFAR10-DVS

Spikformer EMSP 78.53 82.32 Spikformer EMSP w/o DWC 78.17 81.30 Spike-driven Transformer EMSP 79.81 81.10 Spike-driven Transformer EMSP w/o DWC 78.95 80.51 Spikingformer EMSP 79.81 81.95 Spikingformer EMSP w/o DWC 79.44 81.20

D Visualization

D.1 Image Samples Visualization

More dynamic allocation visualizations of the spiking router are given in Fig. 9

E Discussion on the Feasibility of Hardware Deployment for SEMM

SEMM can theoretically be deployed on chips that are synchronous within layers and asynchronous between layers, such as True North [48]. Specifically, multiple spiking router matrices, i.e. {r1, r2, ..., rm}, can be implemented by one block (Linear-Batch Norm-Spiking Neuron), thus can be mapped to the one core of the neuromorphic chip for computation. Outputs from the same core are considered synchronous. The spiking Hadamard product and addition that occur in SEMM are analogous to the AND and ADD operations in SEW Res Net [7]. These element-wise spiking operations can be adapted on neuromorphics chips that support multiple branches and residuals, such as Speck V2 [49].

Figure 9: More visualization samples of routers as masks.

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Justification: Please see the codes in the supplementary material.

Guidelines:

The answer NA means that paper does not include experiments requiring code. Please see the Neur IPS code and data submission guidelines (https://nips.cc/ public/guides/Code Submission Policy) for more details. While we encourage the release of code and data, we understand that this might not be possible, so No is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). The instructions should contain the exact command and environment needed to run to reproduce the results. See the Neur IPS code and data submission guidelines (https: //nips.cc/public/guides/Code Submission Policy) for more details. The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? Answer: [Yes] Justification: Please see Appendix. C. Guidelines:

The answer NA means that the paper does not include experiments. The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. The full details can be provided either with the code, in appendix, or as supplemental material. 7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [No] Guidelines:

The answer NA means that the paper does not include experiments. The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean. It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: Please see Appendix C.2. Guidelines:

The answer NA means that the paper does not include experiments. The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper).

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines?

Answer: [Yes]

Justification: Our work conforms with the Neur IPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: Our work does not involve social impact.

Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our work does not release data or models with a high risk of misuse.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: We describe in Section C.1. Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [NA] Justification: Our work does not introduce new assets. Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [NA] Justification: Our work does not involve crowdsourcing experiments and research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justification: Our work does not study participants. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.