S-TLLR: STDP-inspired Temporal Local Learning Rule for Spiking Neural Networks
Authors: Marco Paul E. Apolinario
Kaushik Roy
Venue: Accepted by TMLR ()

Abstract: Spiking Neural Networks (SNNs) are biologically plausible models that have been identified as potentially apt for deploying energy-efficient intelligence at the edge, particularly for sequential learning tasks. However, training of SNNs poses significant challenges due to the necessity for precise temporal and spatial credit assignment. Back-propagation through time (BPTT) algorithm, whilst the most widely used method for addressing these issues, incurs high computational cost due to its temporal dependency. In this work, we propose S-TLLR, a novel three-factor temporal local learning rule inspired by the Spike-Timing Dependent Plasticity (STDP) mechanism, aimed at training deep SNNs on event-based learning tasks. Furthermore, S-TLLR is designed to have low memory and time complexities, which are independent of the number of time steps, rendering it suitable for online learning on low-power edge devices. To demonstrate the scalability of our proposed method, we have conducted extensive evaluations on event-based datasets spanning a wide range of applications, such as image and gesture recognition, audio classification, and optical flow estimation. S-TLLR achieves comparable accuracy to BPTT (within $\pm2\%$ for most tasks), while reducing memory usage by $5-50\times$ and multiply-accumulate (MAC) operations by $1.3-6.6\times$, particularly when updates are restricted to the last few time-steps.

Published in Transactions on Machine Learning Research (12/2024)
S-TLLR: STDP-inspired Temporal Local Learning Rule for Spiking Neural Networks
Marco P. E. Apolinario
mapolina@ppurdue.c edu
School of Electrical and Computer Engineering Purdue Universily
Kaushik Roy
kaushik Daarauueepuuuee.... edu
School of Electrical and Computer Engineering Purdue University
Reviewed on OpenReview: https: // openreview. net/ per//frum? id= CWas JRcX38
Abstract
Spiking Neural Networks (SNNs) are biologically plausible models that have been identified as potentially apt for deploying energy-efficient intelligence at the edge, particularly for sequential learning lasks. However, training of SNNs poses significant challenges due to the necessity for precise temporal and spatial credit assignment. Back procagapooatiion through time (BPTT) algorithm, whilst the most widely used method for addressing these issues, incurs high computational cost due to its temporal dependency, In this work, we propose S-TLLR, novel three-factor temporal local learning rule inspired by the Spike-Timing Dependent Plasticity (STDP) mechanism, aimed at training deep SNNs learning tasks. Furthermore, S-TLLR is designed to have low memory and time complexities, which are independent of the number of time steps, rendering it suitable for online learning on low-power edge devices. To demonstrate the scalability of our proposed method, we have conducted extensive evaluations on event-based datasets spanning a wide range of applications, such image and gesture recognition, audio classification, and optical flow estimation. S-TLLR achieves comparable accuracy to BPTT (within +2% for most tasks), while reducing memory usage by 50x and multiply-aceummaate (MAC) operations by 1.3 6.6x, particularly when updates are restricted to the last few time-steps.
event-based
1 Introduction
Over the past docade, the field of artificial intelligence has undergone remarkable transformation, driven by a prevalent trend of continuously increasing the size and complexity of neural network models. While this approach has yielded remarkable advancements in various cognitive tasks (Brown et al. 2020 Dosovitskiy
ct al.) 2021). it has come at significant cost: Al systems now demand substantial energy and computational resources. This inherent drawback becomes increasingly apparent when comparing the energy efficiency of current AT systerss with the remarkable efficiency exhibited by the human brain (Roy et al. 2019 Gerstner let al. 2014 Christensen et al.) 2022 Eshraghian et al. 2023). Motivated by this observation, the research community has shown a growing interest in brain-inspired computing. The idea behind this approach is mimic key features of biological neurons, such as spike-based based communication, sparsity, and apatio-temporal processing.
Bio-plausible Spiking Neural Network (SNN) models have emerged as promising avenue in this direction. SNNs have already demonstrated their ability to achieve competitive performance compared to more tra- ditional Artificial Neural Networks (ANNs) while significantly reducing energy consumption per inference
when deployed in the right hardware (Davies et al. 2018 Sengupta al. 2019 Christensen et al. 2022
Code available https:/ http:://hhhhalccccm//mmoooooanao

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Published in Transactions on Machine Learning Research (12/2024)
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Awit]
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BPTT
STDP
S-TLLR
Figure 1: Comparison of weight update computation of a feed-forward spiking layer for BPTT, STDP, and our proposed learning rule S-TLLR. The spiking layer is unrolled over time for the three algorithms while showing the signals involved in the weight updates. The top-down learning signal is shown in green, while the signals locally available to the layer are represented as red for the causal term and blue for non-causal terms, Also, note that the learning signal in BPTT relies on future time steps, whereas in S-TLLR, this signal is computed locally in time.
Neftci et al.) 2019 Roy et al. 2019). One of the main advantages of SNNs lies in their event-driven binary sparse computation and temporal processing based on membrane potential integration. These feature make SNNs well-suited for deploying energy-efficient intelligence at the edge, particularly for sequential learning
tasks (Ponghiran & Roy 2022 Bellee et al. 2020 Christensen et al. 2022
Despite their promise, training SNNs remains challenging due to the necessity of solving precisely temporal and spatial credit assignment problems. While traditional gradient- gradient-based learning algorithms, such as back- propagation through time (BPTT), are highly effective, they incur high computational cost (Eshraghian) et al. 2023 Bellec et al. 2020 Bohnstingl et al. 2022). Specifically, BPTT has memory and time complexity that scales lincarly with the number of time steps (T), such as O(Tn) and O(Tn²), respectively, where n is the number of neurons, making it unsuitable for edge systems where memory and energy budgets are limited. This has motivated several studies that propose learning rules with approximate BPTT gradient computation with constant memory (Bellec et al. 2020 Bohnstingl et al. 2022 Quintana et al. 2023 Ortner
ct al. 2023 Xiao et al. 2022). However, most of those learning rules have a complexity that scales with
the number of synapses (O(n²)) )) as shown in Table D making them expensive for deep convolutional SNN models in practical scenarios (where n > T). Moreover, as most of those learning rules have been derived from BPTT, they only leverage causal relations between the timing of pre- and post-synaptic activities, over- looking non-causal relations used in other learning mechanisms, such as Spike-Timing Dependent Plasticity
(STDP) (Bi & Poo 1998. Song et al. 2000
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Table Comparison of S-TLLR with learning methods from the literature (Where n the number of neurons and T total number of time steps) [Note that complexities are expressed in big O notation for one single layer]
Memory Complexity
Time Complexity
Temporal Local
Leverage Non-Causality
Method
BPTT
Tn n²
Tn²
X
RTRL
Williams & Zipser
1989
n
V
X
e-prop
Bellec et 2020
n²
n!
X
OSTL
Bohnsting
a.. 2022
n²
n12
ETLP
Quintana et Bl.
2023
n²²
n1²
X
OSTTP
Ortner et al.
2023
n²²
n
X
OTTT
Xiao et 2022
n1
X
S-TLLR (Ours)
n1 n1
To overcome the above limitations, in this paper, we propose S-TLLR, novel three-factor temporal local learning rule inspired by the STDP mechanism. Specifically, S-TLLR is designed to train SNNs on event- based learning tasks while incorporating both causal and non-causal relationships between the timing of pre- and post-synaptic activities for updating the synaptic strengths. This feature is inspired by the STDP mechanism from which we propose a generalized parametric STDP equation that uses a secondary activation function to compute the post-synaptic activity. Then, we take this equation to compute an instantaneous eligibility trace [Gerstner et al., 2018) per synapse modulated by third factor in the form a learning signal obtained from the backpropagation of errors through the layers (BP) or by using fixed random fcodback connections directly from the put layer to each hidden layer (Trondheim 2016). Notably, S-TLLR exhibits constant (in time) memory and time complexity, making it well-suited for online learning on resource- constrained devices.
In addition to this, we demonstrate through experimentation that including non-causal information in the learning process results in improved generalization and task performance. Also, we explored S-TLLR in the context of several event-based tasks with different amounts of spatio-temporal information, such as image and gesture recognition, audio classification, and optical flow estimation. For all such tasks, S-TLLR can achieve performance comparable to BPTT and other learning rules, with significantly lower memory and computation requirements.
The main contributions of this work can be summarized as:
We introduce novel temporal local learning rule, S-TLLR, for spiking neural networks, drawing inspiration from the STDP mechanism, while ensuring memory complexity that scales linearly with the number of neurons and remains constant over time.
Demonstrate through experimentation the benefits of considering non-causal relationships in the learning process of spiking neural networks, leading to improved generalization and task performance.
Validate the effectiveness of the proposed learning rule across a diverse range of network topologies, including VGG, RosNet, U-Net-like, and recurrent architectures.
Investigate the applicability of S-TLLR in various event-based camera applications, such as image and gesture recognition, audio classification, and optical flow estimation, broadening the scope of its potential uses.
2 Background
2.1 Spiking Neural Networks (SNNs)
To model the neuronal dynamics of biological neurons, we use the leaky integrate and fire (LIF). The LIF model can be mathematically represented as follows:
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us ((sst 1] 0th9i 1]) myaa((
(1)
y = ((u[t)] vtt)
(2)
where u, represents the membrane potential of the i-th neuron, and wij is the forward synaptic strength between the i-th post-synaptic neuron and the j-th pre-synaptic neuron. Additionally, 7 is the leak factor that reduces the membrane potential over time, Uth is the threshold voltage, and θ is the Heaviside function. Thus, when u, reaches the Uth: the neuron produces output binary spike (v.). This output spike triggers the reset mechanism, represented by the reset signal vra3 [t], which reduces the magnitude of u,
2.2 Spike-Timing Dependent Plasticity (STDP)
STDP is a learning mechanism observed in various neural systems, from invertebrates to mammals, and is believed to play a critical role in the formation and modification of neural connections in the brain in processes such learning and memory |Gerstner et. al. 2014). STDP describes how the synaplic strength (wyy) between two neurons change based on the temporal order of their spiking activity. Specifically, STDP describes the phenomenon by which Wij is potentiated if the pre-synaptic neuron fires just before the post-synaptic neuron fires, and Wy is depressed if the pre-synaptic neuron fires just after the post-synaptic neuron fires. This means that STDP rewards causality and punishes non-causality. However, [Anisimvaa et al. 20222) suggests that STDP favoring causality can be a transitory effect, and over time, STDP evolves to reward both causal and non-causal relations in favor of synchrony. Such general dynamics of STDP can be described by the following equation:
{ apre aaprevire ti-ti 1/-ti apose post
if t. 2 fy (causal term) if t < t (non-causal term)
((4, (4)
where (11;,g) represents the magnitude of the change the synaptic strength (Awsy), 4 and ty represent the firing times of the post- and pre-synaptic neurons, Oprre and Apre are strength and exponential decay factor of the causal term, respectively. Similarly, apost and Apost parameterize the non-causal effect. Note that when apost 0, STDP favors causality, whereas apost 0, STDP favors synchrony.
Based on STDP, the change in the synaptic strengths at time f can be computed forward in time using local variables by the following learning rule (Gerstner et al. 2014):
Awyly = (preWlll
Σ
Nome azy [1'] + Ppot [4]
(4)
P==0
wig + 1] = Wiyl [t] + Awild
(5)
where ([[]] and æy are binary values representing the existence of post- and pre-synaptic activity (spikes) at time t, respectively. Note that both summations in (4) can be computed forward time as a recurrent
equation of the form tr(z,){t tr 1] + x, where (x(;))[t] is trace of æy. Hence, (4) can be
expressed as:
Awylt apre vn (t) + Apost a (t)(() (t(t))
(6)
2.3 BPTT and three-factor (3F) learning rules
BPTT is the algorithm by default used to train spiking neural networks (SNNs) as it is able to solve spatial and temporal credit. assignment problems. BPTT calculates the gradients by unfolding all layers of the network in time and applying the chain rule to compute the gradient as:
dC = Σ Tyll 0u([] dw dw
T ac dylt Ou[t]
(7)
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Although BPTT can yield satisfactory outcomes, its computational requirements scale with time, posing limitation. Moreover, it is widely acknowledged that BPTT is not a biologically plausible method, as highlighted in Lillicrap & Santoro (2019).
In contrast, 3F learning rules Gerstner et al. 2018 are a more biologically plausible method that uses the combination of inputs, outputs, and a top-down learning signal to compute the synaptic plasticity. The general idea of the 3F rules based on that synapses are updated only if a signal called eligibility trace eij is present. This cligibility trace is computed based general functions of the pre- and post-synaptic activity decaying over time. Such behavior is modeled in a general sense on the following recurent equation:
ex Besy [t- 1] + I [v: (1)(())((1)111)))
(8)
Here is an exponential docay factor, (()(t) and g(>,(t)) are element-wise functions of the post- and pre- synaptic activily, respectivelly. Then, the change of the synaptic strengths (waj) is obtained by modulating Eij with top-down learning signal (6:) as:
Awij Σ 8:11esy14
(9)
3F learning rules have demonstrated their effectiveness in training SNNs, as shown by Bellec el al. (2020)
Additionally, it is possible to approximate BPTT using a 3F rule when the learning signal (o:[t) is computed
ac(]] (instantaneous error learning signal), and the eligibility trace approximates Zu tue as discussed as 2y
dy[[] du(d]
by Bellec et al. (2020) and Martin-Sánchez ct al. (2022). However, it is important to note that 3F rules using
eligibility Lraces lormulated as {S) exhibit a memory complexity of On²)), rendering them very expensive in terms of memory requirements for deep convolutional SNNs.
3
Related Work
3.1
Existing methods for training SNNs
Several approaches to train SNNs have been proposed in the literature. In this work, we focus on surrogate
gradients(Neftcii et al.] 2019; Li et al. 2021), and bio-inspired learning rules (Diehl & Cook 2015
Thiele
et al.
2018 Kheradpisheh et al. 2018 Bellec et al. 2020).
Training SNNs based surrogate gradient methods (Neftci et al. 2019
Li et al. 2021 extends the tradi-
tional backpropagation through time (BPTT) algorithm to the domain of SNNs, where the non- non-differentiable firing function is approximated by a continuous function during the backward pass to allow the propagation of crrors. The advantage of these methods is that they can exploit the temporal information of individual spikes so that they can be applied to a broader range of problems than just image classification (Paredes- Vallés & de Croon, 2021 CCamer et al.) (002).. Moreover, such methods can result in models with low latency for energy-efficient inference (Fang al. 2021a). However, training SNNs based on surrogate gradi- ents with BPTT incurs high computational and memory costs because BPTT's requirements scale linearly with the number of time steps. Hence, such methods can not be used for training online under the hardware constraints imposed by edge devices (Neftci et al.) 2019).
Another interesting avenue is the use of bio-inspired learning methods based on the principles of synaptic plasticity observed in biological systems, such as STDP (Bi & Poo. 1998. Song et al.] 2000) or eligibility traces (Gerstner al., 2014 2018), which strengthens or weakens the synaptic connections based on the relative timing of pre- and post-synaptic spikes optionally modulated by top-down learning signal. STDP methods are attractive for on-device learning as they do not require any external supervision or error signal. However, they also have several limitations, such as the need for a large number of training examples and the difficulty of training deep networks or complex ML problems (Diehl & Cook 2015). In contrast, three- factor learning rules using eligibility traces (neuron local synaptic activity) modulated by an error signal, like o-prop [Bellee et al. 2020), can produce more robust learning overcoming limitations of unsupervisod methods such as STDP. Nevertheless, such methods' time and space complexity typically make them too costly to be used in deep SNNs (especially in deep convolutional SNNs).
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3.2 Learning rules addressing the temporal dependency problem
As discussed in the previous section, the training methods based on surrogate gradient using BPTT results in hegh-performance models. However, their major limitations are associated with high computational re- quirements that are unsuitable for low-power devices. Such limitations come from the fact that BPTT has to store copy of all the spiking activity to exploit the temporal dependency of the data during training. Tn order to address the temporal dependency problem, several methods have been proposed where the compu- tational requirements are time-independent while achieving high performance. For instance, the Real-Time Time Recurrent Learning (RTRL) (Williams & Zipser 1989) algorithm can compute exact gradients without the cost of storing all the intermediate states. Although it has not been originally proposed to be used on SNNs, it could be applied to them by combining with surrogate gradients (Neftci et al. 2019). More recently, other methods such as e-prop (Bellec et al.) 2020), OSTL (Bohnstingl et al.) |022), and OTTT (Xiao et al., 2022). derived from BPTT, allows learning on SNNs using only temporally local information (information that can be computed forward in time). However, with the exception of OTTT, all of these methods have memory and time complexities of (n²²) ) or worse, as shown in Table [ making them significantly more expensive than BPTT, when used for deep SNNs in practical scenarios (where n >> T)) Moreover, since those methods (with the exception of RTRL) have been derived as approximations of BPTT, they only use causal relations in the timing between pre- and post- synaptic activity, leaving non-causal relations (as those used in STDP shown in Fig. ! unexplored.
3.3 Combining STDP and backpropagation
As previously discussed, STDP has been used to train SNNs models in an unsupervised manner (Dichl & Cook 20155] Thiele et al.] 2018 Kheradpisheh el al.] 2018). However, such approaches sulfer from severe drawbacks, such as requiring a high number of timesteps (latency), resulting in low accuracy performance and being unable to scale for deep SNNs. So, to overcome such limitations, there have been some previous efforts to use STDP in combination with backpropagation for training SNNs, by either using STDP followed for fine-tuning with BPTT (Tee et al. 2018) or modulating STDP with an error signal Tavannei & Maida 2019: Hu et al. 2017 Hao et al. 20200. However, such methods either do not address the temporal dependency
problem of BPTT or do not scale for deep SNNs or complex computer vision problems.
4
STDP-inspired Temporal Local Learning Rule (S-TLLR)
4.1 Overview of S-TLLR and its key features
We propose a novel 3F learning rule, S-TLLR, which is inspired by the STDP mechanism discussed in Section 2.2 S-TLLR is characterized by its temporally local nature, leveraging non-causal relations in the timing of spiking activity while maintaining low memory complexity Onn).
Regarding memory complexity, conventional 3F learning rule requires an eligibility trace (ey) which involves recurrent equation as described in 18) In such formulation, the eij is state requiring memory that scales linearly with the number of synapses (O(n²)). For the S-TLLR, we dropped the recurrent term and considered only the instantancous term (i.e. 3 == in 8 Instead of requiring 0(n2) memory to store the state of egy, we need to keep track of only two variables (f(v)) and g(x,)) with O(n) memory. Hence, eig(t) can be computed as the right-hand side of (6) which exhibits a memory complexity O(n). This low-memory complexity is key aspect of S-TLLR since it enables the method to be used in deep neural models where
methods such as Williams & Zipser 1989);; ; [Blllcc et al (2020) Bohnstingl et al. (2022) Ortner et al. (0023)
Quintana et al.
2023) are considerably more resource-intent intensive.
Finally, since BPTT is based on the propagation of errors in time, it only uses causal relations to compute gradients, that is, the relation between an output y/t] and previous inputs æft). z[t - .1]...., z00). These causal relations are shown in Fig. (rod dottod line). Also, methods derived from BPTT (Bellee et al.,
2020 Bohnstingl et al. 2022 Ortner et al. 2023 Quintana et al.) 2023 Xiao et al. (202)) use exclusively causal relations. In contrast, we took inspiration from the STDP mechanisms, which use both causal and
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non-causal relations in the spike-timing (Fig. | red and blue dotted lines), to formulate S-TLLR 3F learning rule with learning signal modulating instantaneous eligibility trace signal as shown in Fig. 1
4.2 Technical details and implementation of S-TLLR
As discussed in the previous section, our proposed method, S-TLLR, has the form of three-factor learning rule, Amalt:] 5,ftle; [f], involving top-down learning signal [t and cligibility trace, etylt]. To compute etyl!] we use a generalized version of the STDP equation described in 4 that can use secondary activalion function to compute the postsynaptic activity.
e = apre (u,(t))
E P=0
Xpré 2zsl"] + apost post
( (t,(tt)
(10)
Here, V is secondary activation function that can differ from the firing function used in (2). We found empirically that using a function (u() with f (())du S 1 yields improved results, specific v functions are shown in Appendix A.3 Furthermore, (10p considers both causal (first term on the right side) and non- causal (second term) relations between the timing of post- and pre-synaptic activity, which are not captured in BPTT (or its approximations [Belloe et al.] 2020 Bohnstingl ct al., 2022 Xiao et al., 2022 ) The causal (non-causal) relations are captured as the correlation in the timing between the current post- (pre-) synaptic activity and the low-pass filtered pre- (post-) synaptic activity. Note that (10) can be computed forward in
time (expressing it in the form of 6 and using only information locally available to the neuron as follows:
Esy [1] Opre ((u + (y
V((((())
(11)
The learning signal, 6.(t), is computed the instantaneous error pack-propagated from the top layer (L) the layer (I), as shown in 12 and the synaptic update is done shown in 13
ocly" By('|1
i Ti otherwise
8 [
(12)
Wi == wij + P [t]eet
(13)
Here, P is the learning rate, is the total number of time steps for the forward pass, Tj the initial time step for which the learning signal is available, y* is the ground truth label vector, [ is the output vector of layer L at time t, and C is the loss function. Note that the backpropagation occurs through the layers and not in time, so the S-TLLR is temporally local. Depending on the task, good performance can be achieved even if the learning signal is available just for the last time step TT)
While our primary focus is on proo-bbackpropggation to generate the learning signal, it is worth noting that employing random feedback connections, such as direct feedback aligument (DFA) (Trondheim, 2016 Frenkel et al. 2021), for this purpose is also feasible. In such cases, S-TLLR also exhibits spatial locality. We present some experiments in this direction in Appedix C.1 The S-TLLR algorithm for multilayer implementation
is shown in Algorithm i
4.2.1 S-TLLR for models with recurrent synaptic connections
A more general model of the LIF model includes explicit recurrent connections. Similar to the equation the LIF model featuring explicit recurrent connections can be mathematically represented as:
u (u,(t - 1] Vtayist 1)) + w uff=z1( W Uk
1]
(14)
y ( ( v4t)
(15)

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Here, ue denotes the membrane potential of the i-th neuron. Morcover, wnij represents the forward synaptic connection between the i-th post-synaptic neuron and the j-th pre-synaptic neuron, while w signifies the recurrent synaptic connection between the i-th and k-th neurons in the same layer. Other terms remain consistent with the description in (
The eligibility trace for the forward connections computing following 10 however, since for the recurrent
synaptic connections, the neuron outputs from a previous time step (t 1) serve inputs for the current time step (t) the eligibility trace change to encapsulated this behavior as follows:
c ei [t] Opre (u.(t)
Xpré yult' 1] + Apost 1] _
Xpost (uilt()
(16)
t==0
Note that for models with explicit recurrent connections, memory requirements remain constant, thereby upholding temporal locality.
4.3 Computational improvements
Here we analyzed the theoretical computational improvements in terms of the number of multiply-accumulate (MAC) operations and memory requirements of S-TLLR with respect to BPTT, First, we started by ex- panding the BPTT terms by replacing [ on 7 Assuming an SNN model with L layers and N ncurons we obtain the following
per layer, and appropriately factorizing the terms as described in Appendix B.1 expression for the gradients on the synaptic connections in layer
d.C
T
ac
Σ F=2
(' 1 1T
14
(17)
Oy
1
1
From (34, it can be observed that BPTT is not local in time the updated at time step t' depend future time steps, as illustrated in Fig. 1 Therefore, the BPTT requires information on all the time steps, and consequently, its memory requirements scale linearly with the total number of time steps in the input sequence (T). Then, the total memory required is estimated as follows:
Меmaртт = x Σ N
(18)
To estimate the number of operations, specifically MAC operations, we exclude any element-wise operations. Referring to (34), we ascertain that the number of operations depends on both the number of inputs and outputs, resulting in a total of ((O) ((111) operations. Additionally, we need to account for the operations involved in propagating the learning signals to the previous layer, equating to NO), x ((1-1), Consequently, the estimated number of operations be calculated as follows:
MACBPTT 2T x NO x N(-1)
(19)
A similar analysis is done for S-TLLR. Based on the temporal locality of the method expressed in can conclude that the memory requirements of S-TLLR be computed follows:
Mems-TLLR 2 Σ 20
(20)
Here, the factor 2 is produced by the trace variables required to maintain the temporal information. For the number MACs, we first must note that S-TLLR only updates the weights when the learning signal is present at time Ti, that is weights are updated T - Ti times. Additionally, we account for the operations required by the non-causal terms. Therefore, the number of operations for S-TLLR is:
MACS-TLLR = 3( - Ti) x v(0) x ((1-1)
(21)
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Based on the previous discussion, the theoretical improvements in memory (Smem) and mumber of operations (SMAC) can be obtained as follows:
MemBPTT
T 2i=0 NO
T
Smem
(22)
MemS-TLLR
2 x > 21=0 N
SMAC МАСBPТT 27 E - NO N
2T
(23)
MACs-TLLR 3(T Ti) E NO NO NO-1) 3(T -
It important note that these theoretical improvements do not account for the overhead introduced by specific hardware implementations, which may result in discrepancies between the theoretical and experi- mental results.
A Algorithm S-TLLR algorithm
Require: 2: " w, T 1: for t 11-1,22, 1......,7 T do
2: for 1or111,5 1,2,...., do
Update membrane potential " with (1)
Produce output spikes (t) with (2)
Update eligibility trace ((0 with
6: end for
7:
if 2 Ti then
8: Initialize 8¹ = V. y Cly (ty y*)
Initialize (L)
10:
for I L - 1, ... do
2
11:
if learning signal produced by backpropagation then 8()) = (1+1) (w+1T (0((+)T8(++))) g(++))
12:
(1) 800 o (1) {0' surrogate gradient)
13:
14:
else if learning signal produced by random feedback then
((4) (()) (B() is fixed random matrix)
15:
16:
end if
17:
end for
Aw (L)
[t] = 0 (L- (L-1) (Last layer]
18:
19:
Aw iy t] = o
ejj
(1 (11)) (Hidden layers]
20: end
21: end for
22: for | 1,2, L do
23: Update weights w (1)) + p ET [1
24: end for
25: return wn
5
Experimental Evaluation
5.1 Effects of non-causal terms on learning
We performed ablation studies on the DVS Gesture, DVS CIFARIO, N-CALTECHI01, and SHD datasets to evaluate the effect of the non-causal factor (Apost) on the learning process.
For this purposc, we train VGG9 model, described in Appodix A five times with the same random seods for 30, 30, and 300 epochs in the DVS Gesture, N-CALTECHI01, and DVS CIFARIO datasets, respectively. Similarly, a recurrent SNN (RSNN) model, described in Appedix was trained five times during 200 epochs
on the SHD. To analyze the effect of the non-causal term, we evaluate three values of apost: -1, 0, and
According (10., when apost = 0, only causal terms are considered, while apost (Apost 1) means
that the non-causal term added positively (negatively). As shown in Table 2 for vision tasks, can be

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Published in Transactions on Machine Learning Research (12/2024)
Table 2: Effects of including the non-causal terms in the eligibility traces during learning [Accuracy (st) reported over trials)
Dataset
Model
T
T
(Apost, Apre: Apro)
Opost 0
Apost +1
Apust = -1
DVS Gesture
VGG9
20
15
(0.2, 0.75, 1)
94.61 = 0.73%
94.01 + 1.10%
95.07 = 0.48%
DVS CIFARIO
VGG9
10
5
(0.2, 0.5, 1)
72.93 = 0.94%
73.42 + 0.50%
73.93 1 0.62%
N-CALTECHI01
VGG9
10
5
(0.2, 0.5, 1)
62.24 = 1.22%
53.42 + 1.50%
66.33 = 0.86%
SHD
RSNN
100
10
(0.5, 1, 1)
77.09 1 0.33%
78.23 1 1.84%
74.69 1 0.47%
seen that using Opost = 11 improves the average accuracy performance of the model with respect to only using causal terms Opost 0. In contrast, for SHD, using apost = 1 improves the average performance over using only causal terms, as shown in Table 2 This indicates that considering the non-causal relations of the spiking activity (either positively or negatively) in the learning rule helps to improve the network performance. An explanation for this effect is that the non-causal term acts as a regularization term that allows better exploration of the weights space. Additional ablation studies are presented in Appedix that support the improvements due to the non-causal terms.
5.2 Performance comparison
5.2.1 Image and Gesture Recognition
We train VGG-9 and ResNet18 models during 300 epochs using the Adam optimizer with a learning rate of 0.001. The models were trained five times with different random seeds. The baseline was set using BPTT, while the models trained using S-TLLR used the following STDP parameters (Apost, Apre; Opostt Opre): 00.2,0.75, 1),1) for DVS Gesture and (0.2,0.5, 11, for DVS CIFAR10 and N-CALTECHI01.
The test accuracies are shown in Table 3 In all those tasks, S-TLLR shows a competitive performance compared to the BPTT baseline. In fact, for DVS Gesture and N-CALTECH101, S-TLLR outperforms the average accuracy obtained by the baseline trained with BPTT. Because of the small size of the DVS Gesture dataset and the complexity of the BPTT algorithm, the model overfits quickly resulting in lower performance. In contrast, S-TLLR avoids such overfitting effect due to its simple formulation and by updating the weights only on the last five timesteps. Table 3 also includes results from previous works using spiking models on the same datasets. For DVS Gesture, it can be seen that S-TLLR outperforms previous methods such as Xiao et al. (202)): Shrestha & Orchard (0018;; Kaiser et al. (0200), in some cases with significantly less number of time-steps. In the case of DVS CIFARIO, S-TLLR demonstrates superior performance compared to the baseline with BPTT when the learning signal is utilized across all time steps (T) = 0). Furthermore, S-TLLR surpasses the outcomes presented in Zheng et al. (2021); Fang et al. (2021b) Li al. (2021), yet it lags behind others such Xiao et al. (2022) Deng al. (2022); Meng et al. (2022). However, studics such as Deng et. al. (0022] Meng et al. (2022) showcase exceptional results primarily locused on static lasks without addressing temporal locality or memory efficiency during SNN training. Consequently, although serving as a reference, they do not fairly compare to S-TLLR. The most pertinent comparison lies with Xiao et al. 2022), which shares similar memory and time complexity with S-TLLR. Notably, S-TLLR (T) = O, Oposst 1) exhibits a performance deficit of 0.67%. This difference is primarily attributed to the difference in batch size during training. While Xiao et al. (2022) uses batch size of 128, we were constrained to 48 due to hardware limitations. To validate this point, we trained another model utilizing only causal terms: S-TLLR (Ti = 0, apost 0), equating it to Xiao et al. 2022) considering the selection of STDP parameters and secondary activation function (T) in DVS CIFAR10 experiments. The comparison reveals that S- T-LL (Aiysst = 1) outperforms S-TLLR (Apost = 0) (equivalent to Xiao et al. (2022)) under the same conditions, further corroborating the advantages of including non-causal terms (Aipeet + 0) during training. Finally, when compared to BPTT, S-TLLR significs a 5x memory reduction. By using the lcarning signal for the last five time steps, it effectively diminishes the number of accumulate (MAC) operations by 2.6x for DVS Gesture, and by 1.3x for DVS CIFAR10 and N-CALTECH1O1.
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Table 3: Accuracy performance of methods on vision and audio datasets
Time-steps
Batch Size
Accuracy (mean
# MAC'
Memory'
Method
Model
(T)
(x10")
(MB)
DVS CIFARIO
BPTT
Zheng el.
2021
ResNet-19
67 67.8%
BPTT
Fang cb al.
2021b
PLIF (7 layers)
20
74.8%
TET
Deng ct al. 2022
VGG-11
10
128
83.17 0.15%
DSR
Meng cl al 2022
VGG-11
10
128
77.27 1: 0.24%
BPTT
Li et al.
2021
ResNet-18
10
75.4 0.05%
OTTTA Xiao
2022
VGG-9
76.27 0.05%
BPTT (bascline)
VGG-9
75.44 0.76%
6.82
18.12
S-TLLR (Ours, 7) 5, Opost =
VGG-9
10
73.93 + 0.62%
5.12
3.62
S-TLLR (Ours, T: 0, apout
=-1
VGG-9
10
48
75.6 = 0.10%
10.26
3.62
S-TLLR (Ours, T) 0, Oport =
VGG-9
10
48
74.8 + 0.15%
6.82
3.62
BPTT (baseline)
ResNet18
10
48
T2.68 = 0.87%
7.13
28.14
S-TLLR (Ours,
ResNet18
71.94 0.73%
5.12
5.62
S-TLLR (Ours,
ResNet18
74.5 0.64%
10.24
5.62
DVS Gesture
SLAYER SShrestaa & Orchard 2018
SNN (8 layers)
300
93.64 0.49%
DECOLLE Kaiser al. 2020
SNN layers)
1800
95.54 0.16%
OTTTA Xiao et al. 2022
VGG-9
20
96.88% 95.58 + 1.08%
BPTT
(baseline)
VGG-9
20
6.06
16.13
S-TLLR (Ours)
VGG-9
20
16
97.72 = 0.38%
2.27
1.61
BPTT (baseline)
ResNet18
20
16
94.92 1 0.38%
6.34
25.03
S-TLLR (Ours)
ResNet18
20
16
94.92 = 0.61%
2.27
2.50
N-CALTECHI01
BPTT
She et al.
2022
SNN (12 layers)
10
71.2%
BPTT
Kim Bl.
2023
VGC-16
5
128
64.40%
BPTT
(baseline)
VGG-9
10
16
65.92 + 0.82%
22.81
20.15
S-TLLR (Ours)
VGG-9
66.058 0.92%
17.22
BPTT (baseline)
ResNet18
60.89 = 0.89%
6.34
31.31
S-TLLR (Ours)
Re
10
61.65 O 0.99%
4.27
6.26
SHD
ETLP
Quintana
2023
ALIF-RSNN
128
74.59 0.44%
OSTTP Ortner et
2023
LIF-RSNN
100
77.33 0.8%
BPTT Bouanane et a
2023
LIF-RSNN
100
128
83.41
BPTT
Cramer et al.
LIF-RSNN
100
83.2 + 1.3
BPTT
(baseline)
LIF-RSNN
100
128
70.57 + 0.96
0.054
0.961
S-TLLReP (Ours)
LIF-RSNN
100
128
78.24
1.84%
0.096
0.019
S- S-TLLRDEA (Ours)
LIF-RSNN
100
74.60
0.52%
0.096
0.019
# MAC and Memory are estimated for batch size of following the equations
23 and
22 described in the Section 4.3
Previous studies' accuracy values are pruvided reported in their respective original papers.
5.2.2 Audio Classification
In order to set a baseline, we train the same RSNN with the same hyperparameters using BPTT for five trials, and with the following STPD parameters (0.5, 1, 1). As shown in Table 3 the model trained with S-TLLR outperforms the baseline trained with BPTT. The result shows the capability S-TLLR to achieve high performance and generalization. One reason why the baseline does not perform well, as suggested Cramer al. (2022), is that RSNN trained with BPTT quickly overfits. This also highlights a nice property of S-TLLR. Since it has simpler formulation than BPTT, it can avoid overfitting, resulting in better generalization. However, note that works such as Cramer al. (2022) Bouanane et al. (2023) achieve better performance after carefully selecting the hyperparameters and using data augmentation techniques. In comparison with such works, our method still shows competitive performance with the advantage of having a 50x reduction in memory.
Furthermore, we compared our results with Quintana et al. 20023 Ortner et al. 2023 which uses the same RSNN network structure with LIF and ALIF (LIF with adaptative threshold) neurons and temporal local learning rules. Table shows that using S-TLLR with BP for the learning signal results in better
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Table 4: Comparison of the Average End-Point Error (AFE) on the MVSEC
Zhu et al.
2018b dataset
[AEE lower is better]
Training Method
OD1 AEE
IF1 AEE
IF2 AEE
IF3 AEE
AEE
Models
Type
Sum
FSFN Oppa -.2 (Ours)
S-TLLR
Spiking
0.50
0.76
1.19
1.00
3.45
FSFN = 0.2 (Ours)
S-TLLR
Spiking
0.54
0.78
1.28
1.09
3.69
FSFN Gpost (Ours)
S-TLLR
Spiking
0.50
0.77
1.25
1.08
3.60
FSFN
(baseline)
BPTT
Spiking 0.45
0.76
1.17
1.02
3.40
Apolinario et al.
2023
BPTT
Spiking 0.51
0.82
1.21
1.07
3.61
Kosta Roy| 20
BPTT
Spiking 0.44
0.79
1.37
1.11
3.78
Hagenaars et al.
12021
BPTT
Spiking 0.45
0.73
1.45
1.17
3.80
Zero prediction
1.08
1.29
2.13
1.88
6.38
performance than those obtained with other temporal local learning rules, with the advantage of having a linear memory complexity instead of squared. Moreover, using DFA to generate the learning signal results in competitive performance with the advantage of being local in both time and space.
5.2.3 Event-based Optical Flow
The optical flow estimation is evaluated using the average endpoint error (AEE) metric that measures the Euclidean distance between the predicted flow (ypred) and ground truth flow (yst) per pixel. For consistency, this metric computed only for pixels containing events (P), similar to Apolinario et al. (2023); Kosta &
Roy 2023
Lee al. (2020); Zhuu al. (2018a 2019 given by the following expression:
1 p Σ
AEE
ly pred yTylle
(24)
For this experiment, we trained a Fully-Spiking FlowNet (FSFN) model, discussed in Appodix A with S- TLLR using the following STDP parameters (Apost, Apre, Opree) = (0.5, 0.8, 1) and d'post = [-0.2, 0.2, 0j. The models were trained during 100 epochs using the Adam optimizer with a learning rate of 0.0002, batch size of 8, and with the learning signal obtained from the photometric loss just for the last time step (Z) = 9). As it is shown in Table 4 the FSEN model trained using S-TLLR with Opost = -0.2 shows a performance close to the baseline implementation trained with BPTT. Although we mainly compared our model with BPTT, take things into perspective, we include results from other previous works. Among the spiking models, our model trained with S-TLLR has the second-best average performance (AEE sum) in comparison with such spiking models of similar architecture and size trained with BPTT (Apolinario et al. 2023 Kosta & Roy 2023 Hagenaars et al.) 2021). The results indicate that our method achieves high performance complex spatio-tempora task, such as optical flow estimation, with 5x less memory and a 6.6x reduction in the number of MAC operations by just updating the model in the last time step.
6 Conclusion
Our proposed learning rule, S-TLLR, can achieve competitive performance in comparison to BPTT on several event-based datasets with the advantage of having a constant memory requirement. In contrast to BPTT (or other temporal learning rules) with higher memory requirements O(Tn) (or O(n))), S-TLLR memory is just proportional to the number of neurons O(n). Moreover, in contrast with previous works that are derived from BPTT approximations, and therefore using only causal relations in the spike timing, S-TLLR explores different direction by leveraging causal and non-causal relations based on a generalized parametric STDP equation. We have experimentally demonstrated on several event-based datasets that including such non- causal relations can improve the SNN performance in comparison with temporal local learning rules using just causal relations. Also, we could observe that tasks where spatial information prodominant, such as DVS CIFAR-10, DVS Gesture, N-CALTECH101, and MVSEC, benefit from causality (Aport =1). In contrast, tasks like SHD, where temporal information is predominant, benefit from synchrony (apost = 1).
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Moreover, by computing the learning signal just for the last few time steps, S-TLLR reduces the number of MAC operations in the range of 1.3x to 6.6x. In summary, S-TLLR can achieve high performance while being mefoer-efficient and requiring only information locally in time, therefore enabling online updates.
Acknowledgments
This work was supported in part by the Center for CoDesign of Cognitive Systems (CoCoSys), one of seven centers in JUMP 2.0, funded by the Semiconductor Research Corporation (SRC) and DARPA, in part by the MiAr program of IARPA, the DoE Microelectronics program, the National Science Foundation, and Intel Corporation.
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A Datasets and experimental setup
We conducted experiments various event-based datasets, including DVS Gesture Amir et al. N-CALTECHI01 (Orchard et al.) 2015 DVS CIFAR-10 (Li et al.] 2017). SHD (Cramer et al. 2022 and MVSEC (Zhu et al.) 2018b). These datasets encompass a wide range of applications, such as image and gesture recognition, audio classification, and optical flow ostimation. In this section, we outline the experimental setup employed in Section 5, covering SNN architectures, dataset preprocessing, and loss functions.
2017
A.1 Network architectures
For experiments on image and gesture recognition, we use a VGG-9 model with the following structure: 64C3- AAA
In this notation, '64C3' signifies convolutional layer with 64 output channels and 3x3 kernel, 'AP2S2' represents average-pooling layers with 2x2 kernel and stride of 2, and 'FC' denotes a fully connected layer. In addition, instead of
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batch normalization, we weight standardization Qiao al. 2019 similar to Xiao et al. (002)). Also, the leak factor and threshold for the LIF models (1) are 7 = 0.5 and = 0.8, respectively.
For experiments with the SHD dataset, we employ recurrent SNN (RSNN) consisting of one recurrent layer with 450 neurons and a leaky integrator readout layer with 20 neurons, following a similar configuration as
in Ortner et al. (2023 Quintana et al. (2023). Both layers are configured with a leak factor of ? = 0.99, and the recurrent LIF layer's threshold voltage is set to Uth 0.8.
We utilize the Fully-Spiking FlowNet (FSFN) for optical flow estimation, as introduced by Apolinario et al. (2023). The FSFN adopts a U-Net-like architecture characterized by performing binary spike computation in all layers. Notably, we enhance the model by incorporating weight standardization (Qiao et al.) 2019) in all convolutional layers. Additionally, our training approach involves ten time sleps without temporal input encoding, in contrast to Apolinaari et al. (2023) that uses the encoding method proposed by Lee et al. (020 Also, the leak factor and threshold of the LIF neurons used for our FSFN are 0.88 and 0.6, respectively.
A.2
Data pre-processing
Here we describe the data pro-processing for cach dataset:
DVS Gesture: the dataset contains 11 hand gesture categorics from 29 subjects under 3 illumination conditions recorded with a Dynamic Vision Sensor (DVS) with resolution of 128x 128 pixels. The recordings were split into sequences of 1.5 seconds of duration, and the events were accumulated into 20 bins (event frames), with each bin having a 75 ms time window. Then, the event frames were resized to a size of 32 x 32, while maintaining the positive and negative polarities as channels.
N-CALTECH: is a spiking version of the original frame-based CALTECH101 dataset. It pro- duced by recording the static images of the CALTECH101 dataset displayed L.CD monitor using an ATIS sensor (an event-based camera) while moving it. The recordings were wrapped into ten bins with the same time window size for each bin (30 ms). Then, the event frames were resized to a dimension of 60 x 45.
DVS CIFAR10: is a spiking version of 10000 samples from the frame-based CIFAR10 dataset recorded with a event-based camera at resolution of 128 x 128 pixels. The recordings were wrapped into ten bins with the same time window size for each bin. Then, the event frames were resized to a dimension of 48 x 48. Additionally, a random crop with a padding of 4 was used for data augmentation.
SHD: is audio-classiffcation dataset consisting of spoken digits ranging from 0 to 9 in both English and German. The audio waveforms were converted into spike trains using an artificial model of the inner ear auditory system. The events (spikes) in each sequence were wrapped into 100 bins, each with a time window duration of 10 ms. No data augmentation techniques were used for SHD.
MVSEC: is event-based dataset used for training and evaluating optical flow predictions. It con- tains stereo event-based data collected in various environments, including indoor flying and outdoor driving, along with the corresponding ground truth optical flow. The events between two consec- ulive grayscale frames were wrapped into ten bins, keeping the negative and positive polarities as channels, Then, the event frames are fed to the SNN model sequentially, similar to the approach in Apolinario et al. (023).
A.3 Loss functions and secondary activation functions (v)
For image, gesture, and audio classification tasks, we utilized cross-entropy (CE) loss and computed the learning signal using 12 with ground truth labels (y*). In contrast, for optical flow, we employed a self-
supervised loss based on photometric and smooth loss, as detailed in Equation (5) in Lee et al. (2020)
Regarding the generation of the learning signal (6): in the context of image and gesture recognition, is exclusively generated for the final five time steps (T) =)). For audio classification, we employ Ti 90, while
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for optical flow, use T 1. This setting reduces the number of computations compared to BPTT by factors of 4x, 1,1x, and 10x, respectively.
Finally, we consider the following secondary activation functions for the computation of the eligibility traces 11:
1
(uu[tt)) (100) uz Uthl + 1)2
(25)
((x[()) = 0.3 max 0. - |ui(0] vthl, (uu[(]) = x sigmoid(ulb)) - v1.) sigmoid(u:lt vra)
(26)
(27)
v (u,(t)) = 1 [t] v2))²
(28)
Here, max(a, b) returns the maximum between a and b, and |- represents the absolute value function. These
activation functions (25). 26), (27 and (88 are specifically used for SHD, DVS Gesture, DVS CIFAR1O,
and MVSEC, respectively.
Note that the secondary activation function plays role similar to that of the surrogate gradient in BPTT, the selected secondary functions were adapted from previous works that utilized similar functions in BPTT
schemes. Specifically, (25) was adapted from Ortner et al. (0023 26 from Apolinario et al. (2023 27
from Wu et al. (2018) and 28 from Hagenaars et al. (2021).
A.4
Experiments Setup
A.4.1 Experimental setup:
effects of non-causal terms learning
For the experiments conducted on DVS Gesture, N-CALTECH101, and DVS CIFAR10 datasets, we trained a VGG9 model for 30, 30, and 300 epochs, respectively. In all these experiments, we utilized the Adam optimizer with a learning rate of 0.001. Additionally, the learning signal was presented only during the last five time steps, meaning the model was updated and the error computed five times per sample. For these experiments, the parameters Apost Xpre: and Opre were kept constant, while Opoost was varied to simulate different scenarios: the absence of non-causal terms (Appet 0), the inclusion of non-causal terms (Apost +1), and the subtraction of non-causal terms (Opost = -1). This approach allowed us to evaluate the impact of non-causal terms on the learning rule.
For the experiments on the SHD dataset, we trained a recurrent spiking neural network (SNN) model for 200 epochs. this setup, we used the Adam optimizer with lcarning rate of 0.0002 and a batch size of 128. Similar the previous experiments, O'post was varied among three values (-1, 0, and +1) to assess the effect of non-causal terms while keeping all other parameters constant.
B Computational analysis of BPTT and S-TLLR
we discussed the computational improvements of S-TLLR over BPTT. Here, we further detailed the procodure for obtaining (34), and present an example of real GPU memory usage with BPTT and S-T.RR
Section
1.3
B.1 BPTT analysis
To analyze BPTT, we follow a similar analysis as Bellec et al. (2020). Here, we will utilize a threc-layer feedforward SNN as illustrated in Fig. B Our analysis based on a regression problem with the target denoted as y* across T time steps, and our objective is to compute the gradients for the weights of the first. layer (w). The Mean Squared Error (MSE) loss function (C) is defined as follows:
T
C =
- "1 2
Σ (3) 1
(29)
t=0
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Myh-1]
Dyht-11
Mij 11
PAN
Ma
Maf
Ma
8C
DC
By
M4
vill 11
M
vul
wxp
Arau(8
3vF-1]
[eft-1]
BPTT
S-TLLR
Figure 2: Comparison of weight update computation of three feed-forward spiking layers for BPTT and S-TLLR. The spiking layers are unrolled over time for both algorithms while showing the signals involved in the weight updates. The top-down learning signal for BPTT (2gf)) is shown in green, note that at time step this learning signal depends on future time steps. In contrast, for S-TLLR the learning signal (1ag llcas) and all the variables involved are computed with information available only the time step t.
8C
Here (*) is the output of the third layer (last layer) at the time step (t). Then, the gradients with respect to (1) can be computed as follows:
dC dw
OC Oul") [+] Ou(Ott Ow(()
(30)
If we expand the term Bu
0C
we obtain the following expression:
OC
OC By (1()
O
Ou (1) (1 +
(31)
Ou (
By (t
Du Ou (1
Ou O
ac Note that the right-hant hand side contains the term du0 that could be further expanded. If we apply 31 recursively, replace it on (30,, and factorize the By
ac
terms, we obtain the following expression:
dc
Oc (t ( W Ou( 1]
Ou k)
(t)
(32)
(1)
Ow((1)
r=t
By doing change of variables between & and 8,, we can rewrite 32 a8:
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dC
T
ac
By Dul
2u - k] Onlli -
a [t]
du
(33)
dw
[y
Ouo(1)
M=
k=U
Further replacing the LIF equation (1) on 33
we obtain:
dC
Oc
('n
0
Σ
7
y 10
(34)
dw(l)
By
1=0
Oc
1y(0i O'
(TD
7 p'
(0)
(35)
The total update of the weight for the first layer is shown in 334 where at each time step, the synaptic updated is represented by (35 Here, it can be seen that the contribution of any time step (f') has two components,
(t(u")())) Ei 7" [t) that time-step t' depends only on previous information (0, ...,,... t' 1, 1,t"). and an learning signal S depends information of future time steps (t' + 1, + 2,.,,, T)- Those components are visualized in Fig. B Since the error signal depends on future time steps can not be computed locally in time, that is with information available only at the time step (t"). Therefore, BPTT is not temporal local learning rule.
B.2 Example of real GPU memory usage
This section shows the substantial memory demands associated with BPTT and their correlation with the nunber of time steps (T). To illustrate this point, we employ simple regression problem utilizing a synthetic dataset (ax, 3²) where x denotes a vector of dimension 1000 and y" represents a scalar value. The batch size used 512. The structure of the SNN model comprises five layers, structured as follows: 1000FC-1000FC- 1000FC-1000FC-I1 In addition, the loss computed solely for the final time step as L (y[") -0]]--"²2, This evaluation conducted on sequences of varying lengths (10, 25, 50, 100, 200, and 300). To simplify, these sequences generated by repeating the same input (z) multiple times. Throughout these experiments, we utilized an NVIDIA GeForee GTX 1060 and recorded the peak memory allocation. The obtained results are visualized in Fig. B.2 These results distinctly highlight how the memory usage of BPTT scales linearly with the number of time steps (T',, while S-TLLR remains constant.
C Additional experiments on the effects of STDP parameters on learning
C.1 Effects of causality and non-causality factors using DFA
In Section 5.1 of the main text, we examined the impact introducing non-causal terms in the computation of the instantaneous eligibility trace (10) when using prror-backproppgatiin (BP) to generate the learning signal. In this section, we conduct similar experiment, but this time, we employ direct foodback alignment (DFA) for the learning signal generation.
As in Section 5.1, we vary the values of apost: including -1, 0, and 1, to assess the effect of non-causal terms. Interestingly, when the learning signal is produced via random feedback with DFA, there is no significant difference observed when including non-causal terms or not. For example, in the RSNN model, using post = 1 yields slightly better performance, as depicted in Fig. C.1 but the difference marginal. Similarly, for the VGG9 model, the performance of apost = 1 and appst
comparable and superior appst 1, as shown in Fig. C.I. This suggests that a more precise learning signal, such BP, may be necessary to fully exploit the benefits of non-causal terms.
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BPTT S-TLLR
3500
3000
GPU memory (MB)
2000
1500
1000
500
0
10
25
50
100
200
300
Timesteps
GPU memory usage for BPTT and S-TLLR for five-layer fully connected SNN models with a different number of time steps (T).
Figure 3:
82
93.0
80
92.5
Accuracy (%)
78
92.0
76
91.5
91.0
74
90.5
72
90.0
70
0
1
-1
1
apsst
apost
SHD
DVS Gesture
Figure 4: Evaluating the effect of Oppost DVS Gesture and SHD datasets using DFA for learning signal generation. Constant STDP parameters for DVS Gesture are (Aposty Apres appre) = (0.2, 0.75, 1), and for SHD, they are (Apnsts Apres prre) = (0.5, 1). The solid purple line represents the median value, and the dashed black line represents the mean value averaged over five trials.
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95
90
Accuracy (%)
85
80
75
70
65
0.25
0.5
0.75
1
Apre
Figure Effects of using a decaying factor, Apres different from the leak spiking parameter (? = 0.5) to compute the causal term on the cligibility trace, with constant parameters (Aposts Oppost Apre) (0.2, 1, when the learning signal generated using BP. Plots based five trials. The solid purple line represents the median value, and the dashed black line represents the mean value.
C.2 Effects of Apre on the learning
In this section, we explore the impact of using Apre value different from the leak parameter 7 of the LIF model (1) on the S-TLLR eligibility trace (10). This parameter, which controls the decay factor of the input trace, offers opportunity to optimize the model's performance. While previous works aiming to approximate BPTT Xiao et al. (2022): Bellec et al. (2020); Bohnstingl et al. (2022) set Xpre equal to the leak factor (7), our experiments, as depicted in Fig. C.1 suggest that using a slightly higher Apre can lead
to improved average accuracy performance.
C.3 Ablation studies on the secondary activation function (7)
In this section, we present ablation studies on the secondary activation function using the DVS Gesture, NCALTECH101, and SHD datascts, with the same configurations used for experiments on Table 2. The results are presented in Table 5 where it can be seen that independent of & function using a non-zero Opost result in better performance than using Post 0.
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Table 5: Ablation studies the secondary activation function v
[Accuracy (mean+std) reported over 5
trials
V function Model
T
Ti
(Apost. Apre apre)
Apost
Opost +1
Opost = =-1
DVS128 Gesture
VGG9
20
(0.2, 0.75, 1) (0.2, 0.75,
73.61 2.92%
81.89 = 5. 5.13%
65.45 + 2.92%
VGG9
20
15
94.61 + 0.73%
94.01 + 1.10%
95.07 0.48%
27
VGG9
20
15
(0.2, 0.75,
94.46 + 0.45%
94.46 + 0.45%
95.85 + 0.66%
NCALTECH101
155
VGG9
10
(0.2, 0.5, 1)
35.11 0.40%
37.19 + 1.17%
28.53
0.56%
VGG9
10
(0.2, 0.5, 1) (0.2, 0.5, 1)
63.34 0.96%
54.07 + 2.37%
66.05 0.92%
VGG9
10
62.24 + 1.22%
53.42 + 1.50%
66.33 0.86%
SHDD
25
RSNN
100
10
(0.5, 1, 1)
77.09 + 0.33%
78.23 + 1.84%
74.69 + 0.47%
RSNN
100
10
(0.5, 1, 1)
76.25 0.44%
76.28 + 0.25%
74.40 0.59%
RSNN
100
(0.5, 1, 1)
75.22 0.79%
76.29 + 0.25%
74.46 0.65%
24


Reviews:
[{"review_id": "EQOrSUBpNe", "reviewer": "Reviewer_7JMW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- the topic of efficient SNNs training is of interest to the audience of this journal\n- the paper is well structured and overall well written, although there's some redundancy in how some concepts (like the computational cost of competing algorithms like backpropagation through time, BPTT) are repeated over and over\n- related work is appropriately cited\n- estimates on number of MAC operations and memory consumption compare favorably against the BPTT baseline, at the expense of limited accuracy degradation. Hence, S-TLLR could find practical application in resource-constraint environments\n\nWeaknesses\n- the main novel contribution of this paper resides in a modification of the eligibility trace formulation to incorporate a secondary activation function in the causal and non-causal terms of eq 10. The authors several choices for activation functions in appendix A.3 and select different functions for different tasks/datasets. It's unclear (and not discussed) what drove this particular selection. There is a limited comparison of outcome in appendix C.3 but how would one go about selecting an activation function for a different task than those described? \n- similarly, the choice of hyperparameters ($T_l$, $\\lambda_{post}$, $\\lambda_{pre}$, $\\cdots$) appears to be strongly task-dependent. What kind of search was performed on the hyperparameter space by the authors? How are performance impacted by the choice of hyperparameters?\n- Memory consumption and MAC ops are theoretical estimates that don't take into account overheads and constraints associated with real hardware. The authors provide one example of actual measurements on GPU in appendix B.3, demonstrating constant memory consumption of S-TLLR with time $T$. Is the measured ratio between BPTT and S-TLLR memory consumption consistent with the estimated one ($T/2$)?\n\nTypos\n- Section 2.2: the term tr$x_j$ is sometimes not set to italics. Please also consider adding parenthesis to the trace function tr($x_j$)\n- Section 2.3: respectivetly $\\rightarrow$ respectively", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces S-TLLR (STDP-inspired Temporal Local Learning Rule), a mechanism to train deep Spike Neural Networks (SNNs) that takes into account pre- (causal) and post- (non-causal) synaptic activity during training. The main advantage of S-TLLR against competing SNNs training strategies is the favorable memory complexity, $O(n)$, which scales linearly with the number of neurons $n$ and is constant in time $T$. The authors provide estimates of number of MAC operations and memory consumption, as well as accuracy/error performance on a collection of tasks for image, gesture, audio classification and optical flow estimation.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "As outlined in the previous section.", "broader_impact_concerns": "No concerns one my end.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "s5I2Uo2hWs", "reviewer": "Reviewer_CiSd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n- The biggest strength of the presented approach is the significant improvements in computational complexity and memory use.\n- I liked the writing and presentation. No major change is needed but please see my comments in the \"requested changes\" section.\n- I did not find any mistake in the computations/algorithm.\n\n### Weaknesses\n- The biggest weakness is that the approach seems a little hand-wavy... More specifically, as far as I understand, it is a combination of BPTT and STDP. While these two are based on clear learning principles, which are standard gradient descent optimization and inspirations from neural learning systems, the presented approach does not come from the first principles. For example,\n  - The impact of setting $\\beta=0$ (from a learning perspective) is not discussed. Is this a simple trick to handle memory inefficiency? Wouldn't breaking the recurrence completely alter the learning signal?\n  - Likewise, what is the role of $\\Psi$? Is it a non-linearity to make the whole framework more expressive or does it play a more crucial role? What would it be its counterpart in BPTT or STDP or alternative approaches?\n- The results don't seem very impressive and are also slightly inconsistent. Is the presented approach now more advantageous in general? What would be a simple principle (e.g., based on the amount and modality of data, difficulty of problem, etc) that would help us choose this approach over others?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work presents a new local learning rule for spiking neural networks. By design, it has low memory and compute complexities. Further, unlike the standard approach (backprop-through-time), the proposed method includes \"non-causal relations\" in its computations. Its effectiveness on event-based datasets is experimentally shown.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- $x_j[t]$ in (1) is not defined near (1).\n- The comment \"$\\gamma$ is the leak factor that reduces the membrane potential over time\" should be elaborated as it's not clear immediately.\n- Based on section 4.2.1, $u_i[t]$ in (10) is not computed recursively. Then, how is it computed?\n- A discussion on $\\Psi$ would be great.\n- About the note \"Note that the backpropagation occurs through the layers and not in time\"; what equation implies that? Is it due to (12) or breaking the recurrence?\n- Highlighting the results in bold would be nice.\n- Ideally, one can compute the memory footprints and the number of floating point operations of all methods (e.g., for one epoch) and add these numbers to a separate column in Table 3. This would let us compare the presented approach and all other baselines for which #MAC and Memory is not reported.", "broader_impact_concerns": "-", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "rcTi2rLQCs", "reviewer": "Reviewer_FrTj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n\n+ Large-scale experiments across different datasets and models.\n+ The proposed method has reduced time and memory complexity.\n\n\nWeaknesses:\n\n- The effectiveness of non-casual terms requires further justification. \n- The performance of S-TLLR does not outperform other methods on 4 datasets.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents S-TLLR, a three-factor temporal local learning rule for spiking neural network training on even-based tasks. Besides the causal relations between the timing of pre- and post-synaptic activities, which is commonly used by existing training methods, the authors extensively take the non-causal relations into account to further enhance the training performance. The evaluation is conducted on various datasets and different tasks, and the experiment results show that the proposed learning method obtains competitive performance compared to BPTT and other learning rules.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The idea of this manuscript is interesting; however, some technical details and experiment results lack clarifications and justifications.  Please see the comments below:\n\n- My major concern is the effects of the non-causal factor. From Table 2, only minor improvements are observed on 4 datasets, and the value of the factor has different impacts on different tasks. To me, the results are not convincing that the non-causal can indeed benefit the S-TLLR. \n\n- The authors may add some brief introductions to the datasets used for in experiments. There is a lack of detailed explanation of the experiment setup, which makes the results somehow hard to interpret. \n\n- The performance of S-TLLR does not outperform BPTT and other methods (Table 3). The author may elaborate more on the advantages and motivations of adapting the proposed method instead of the existing ones.\n\n- The presentation requires further improvement; I noticed several typos and grammar issues (e.g., Section 5.1, non-casual -> non-causal).", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper can provide grounds for further research for a not-that-popular research direction in the community. As the claims are also demonstrated (and considering the requested clarifications to be included) and the novelty is sufficient regarding the criteria of TMLR, I suggest acceptance of the paper after a minor revision.
Decision: Accept (poster)