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REVIEW 3 major objections 5 minor 15 references

Efficient Parallel Training Methods for Spiking Neural Networks with Constant Time Complexity

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By rewriting LIF dynamics as a fixed point over all timesteps, the paper claims SNN training can run in about three parallel iterations instead of T sequential ones, matching BPTT accuracy while leaving the architecture untouched.

desk verdict The fixed-point framework is a genuinely neat unification of PSN/PSU, but the 'constant time' claim doesn't hold up—per-iteration cost grows with T. read the letter →

arxiv 2506.12087 v1 pith:K2W32HTS submitted 2025-06-10 cs.NE cs.AI

Wanjin Feng , Xingyu Gao , Wenqian Du , Hailong Shi , Peilin Zhao , Pengcheng Wu , Chunyan Miao This is my paper · ORCID
classification cs.NEcs.AI MSC 68T0765H10
keywords spikingneuralnetworksfixed-pointiterationparalleltrainingLIFneuronsurrogategradientbackpropagationthroughtimecomplexity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the main bottleneck of spiking-neural-network training — stepping through $T$ timesteps one at a time — can be bypassed by rewriting the Leaky Integrate-and-Fire (LIF) neuron, the standard model whose membrane potential decays between spikes and fires when crossing a threshold, as a single fixed-point equation over the entire time axis. The proposed method, FPT, solves this equation with about three surrogate-smoothed iterations that update every timestep in parallel, and a contraction proof guarantees convergence to the same trajectory the sequential neuron would produce, reset mechanism included. If the claim holds, model architecture and trained weights stay identical to ordinary backpropagation-through-time (BPTT) training, yet measured training time decouples from sequence length: roughly 10 times faster than BPTT at $T=32$ and about 21 times faster at $T=512$, with accuracy statistically indistinguishable from BPTT. Existing parallel spiking neuron (PSN) and parallel spiking unit (PSU) models turn out to be the one- and two-iteration truncations of the same scheme, which explains why they deviate from true LIF dynamics.

What carries the argument

The load-bearing object is the vectorized fixed-point map $\hat\Phi_\alpha(u) = -V_{\mathrm{th}}(\Lambda - I)S_\alpha(u - V_{\mathrm{th}}) + \Lambda c$, built from the lower-triangular Toeplitz decay matrix $\Lambda$ whose $(i,j)$ entry is $\lambda^{i-j}$ for $i \ge j$ and zero otherwise; this matrix encodes how inputs decay into every later timestep. Writing the membrane potential as a $T$-vector $u$ turns the chronological LIF recurrence into a single simultaneous equation, so each application of $\hat\Phi_\alpha$ updates all timesteps in parallel, and the contraction condition $V_{\mathrm{th}} L_\alpha \lVert \Lambda - I \rVert_\infty < 1$, with $\lVert \Lambda - I \rVert_\infty = \lambda(1-\lambda^{T-1})/(1-\lambda)$, guarantees convergence to a unique fixed point. The split steepness parameters, forward $\alpha_f$ and backward $\alpha_b = \alpha_f/3$, keep the forward trajectory close to the true Heaviside dynamics while giving the backward pass a smooth surrogate gradient through $K$ unrolled iterations, avoiding the Jacobian inversion of deep equilibrium models. With a learnable decay matrix $A$ and threshold $B$, the same map yields PSN at $K=1$ and PSU at $K=2$ as truncated special cases.

What would settle it

Run FPT's forward pass with $K=3$ fixed on one GPU and measure per-batch time at $T=8$, $32$, $128$, $512$, and $2048$, holding the network and batch size constant; the paper's own Table 3 places the curve at 1.55 ms ($T=8$), 2.01 ms ($T=32$), and 5.58 ms ($T=128$), so a rising curve — or a direct operation count showing the $\Lambda c$ product grows with $T$ — would settle that the complexity is not constant in $T$ as literally claimed.

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Extended reading notes

Core claim

FPT's central discovery is that the LIF dynamics $u_t = \lambda(u_{t-1} - V_{\mathrm{th}} s_{t-1}) + c_t$, $s_t = H(u_t - V_{\mathrm{th}})$ across $T$ timesteps can be re-expressed exactly as one simultaneous system in vector form: $u = -V_{\mathrm{th}}(\Lambda - I)s + \Lambda c$, $s = H(u - V_{\mathrm{th}})$, where $\Lambda$ is the lower-triangular matrix of exponential decay weights $\lambda^{i-j}$. Replacing the Heaviside step $H$ with a sigmoid surrogate $S_\alpha$ makes the induced map $\hat\Phi_\alpha(u) = -V_{\mathrm{th}}(\Lambda - I)S_\alpha(u - V_{\mathrm{th}}) + \Lambda c$ a contraction under the 1-norm whenever $V_{\mathrm{th}} L_\alpha \lambda(1-\lambda^{T-1})/(1-\lambda) < 1$, so $K$ iterations — $K=3$ in practice — reach a unique fixed point whose spike outputs reproduce the original neuron's trajectory. Splitting the surrogate steepness, with a large $\alpha_f$ in the forward pass and a smoother $\alpha_b = \alpha_f/3$ in the backward pass, keeps the forward map faithful while stabilizing gradients, and training needs no architecture change. The paper further shows that PSN ($K=1$) and PSU ($K=2$) are special cases of this same iterated map for a learnable variant, and that the preserved reset mechanism — absent from those models — is what keeps FPT accuracy at LIF levels. The claimed time complexity is $O(K)$, versus $O(T)$ for BPTT, because each iteration updates all $T$ timesteps simultaneously.

Load-bearing premise

The paper's constant-time claim counts each full pass over all $T$ timesteps as a single unit of work, even though that pass multiplies by the $T$-by-$T$ decay matrix $\Lambda$; the cost of forming $\Lambda c$ and $(\Lambda - I)s$ is never analyzed, so the $O(K)$ headline holds only if that matrix-vector cost is ignored.

Editorial extensions

If this is right

  • Training cost decouples from sequence length: FPT reports per-batch training about 10 times faster than BPTT at $T=32$–$128$ and about 21 times faster at $T=512$, with inference time nearly constant in $T$.
  • Accuracy matches full BPTT: t-tests on the Amazon Photos/DRSGNN benchmark give $p > 0.05$ at $T=8$, $32$, and $128$, while FPT beats the LocalZO baseline by over 3 accuracy points on DVS-CIFAR10 and by about 1.7 points on ImageNet-100.
  • Because the converged forward dynamics equal the original LIF dynamics, a model trained with FPT can be deployed unchanged on sequential neuromorphic hardware, and the paper argues this makes FPT a practical pretraining scheme for long-sequence SNN tasks.
  • PSN and PSU are revealed as one- and two-step truncations of a single iterated map; their known accuracy gap relative to LIF is attributable to the missing iterations and the dropped reset mechanism, both of which FPT retains.
  • Three iterations suffice across all reported settings: increasing to $K=4$ or $K=5$ leaves DVS-Gesture accuracy unchanged, and non-learnable FPT with $K=3$ reproduces ordinary LIF accuracy on sequential CIFAR10 (81.54 vs 81.50).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A literal reading of the $O(K)$ claim counts only the number of iteration rounds: each round computes the $T$-vector $-V_{\mathrm{th}}(\Lambda - I)s + \Lambda c$, and producing any $T$-dimensional output costs $\Omega(T)$ work — $O(T)$ work and $O(\log T)$ parallel depth with a scan, or $O(T \log T)$ with an FFT — so the honest complexity is closer to $O(K \log T)$ parallel depth and $\Omega(KT)$
  • Because $\Lambda$ is Toeplitz, $\Lambda c$ is a causal exponential filter — effectively a parallel prefix or convolution — so FPT's remaining $T$-dependence can be delegated to a purpose-built scan or FFT kernel; the method's true complexity would then be that of the convolution, not of the fixed-point iterations.
  • Since the fixed point is unique and contractive, a deep-equilibrium-style rule that differentiates the equilibrium equation directly is a natural alternative to unrolling $K$ iterations; the paper chooses the unroll because $K$ is small, and that trade-off would flip for larger $K$, steeper $\alpha_f$, or cheaper Jacobian solvers.
  • The same vectorization should transfer to any contractive leaky recurrence $u_t = \lambda g(u_{t-1}) + c_t$ with a Lipschitz $g$; the paper treats only the LIF/Heaviside pair, but the fixed-point template suggests a route to parallelizing other gated recurrent cells that have a reset.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper proposes Fixed-point Parallel Training (FPT), a method that rewrites the LIF recurrence over T timesteps as a fixed-point equation u = -Vth(Λ-I)s + Λc with s = S_α(u - Vth), and solves it by K fixed-point iterations so that all timesteps can be updated simultaneously. The backward pass backpropagates through the K iterations using surrogate gradients. The authors claim O(K) training time, prove convergence of the surrogate fixed-point map, identify PSN and PSU as K=1 and K=2 special cases, and report competitive accuracy and large GPU speedups on DVS-CIFAR10, DVS-Gesture, ImageNet-100, sequential CIFAR10/CIFAR100, and an Amazon Photos graph task.

Significance. If the headline complexity claim were supported, FPT would be a valuable architecture-agnostic SNN training method because it preserves the reset mechanism and can be applied without modifying the network. The empirical part is careful in several respects: results are averaged over three runs with standard deviations, t-tests are used against BPTT, and speedups are measured on actual GPUs. The theoretical connection to PSN and PSU is also a useful conceptual contribution. However, the central O(K) time-complexity claim is not backed by a cost analysis of the T-dimensional matrix operations, and the paper's own Table 3 shows a clear dependence of FPT training time on T. With a corrected complexity statement, the contribution remains solid, but as written the abstract, title, and Section 5.3 overstate what is established.

major comments (3)
  1. [§4.3.1, Eq. (9); Algorithm 1; §5.3, Table 1] The claim that FPT has O(K) time complexity is not supported. Each iteration computes u^{(k)} = -Vth(Λ-I)s^{(k-1)} + Λc, where Λ is a T×T lower-triangular Toeplitz matrix defined in Eq. (8). Applying Λc or (Λ-I)s costs Ω(T^2) with a dense matvec, or Ω(T) work and O(log T) depth with a prefix scan. The term (Λ-I)s must be recomputed in every iteration because s changes, so the per-iteration cost is not O(1); the total cost is Ω(LKT) work for an L-layer network, not O(LK). The paper's own Table 3 is consistent with a T-dependent cost: with K=3 fixed, FPT training time increases from 1.55 ms at T=8 to 2.01 ms at T=32 and 5.58 ms at T=128. The abstract's and title's 'constant time complexity' claim therefore needs to be revised, or a cost model for the T-dimensional operations must be provided.
  2. [§5.1, Lemma 5.1; Section 6.1, Figure 2] Lemma 5.1 proves contraction of the surrogate map Φ̂_α(u) = -Vth(Λ-I)S_α(u-Vth)+Λc, but it does not bound the distance between the fixed point of Φ̂_α and the fixed point of the original LIF map with the Heaviside step. The statement that FPT 'preserves LIF dynamics' is therefore not established by the theory alone; Figure 2 provides empirical evidence for specific α and K values. A quantitative approximation bound, such as ||u*_α - u*_LIF|| as a function of α, λ, and T, would be needed to substantiate the theoretical claim, especially since the schedules in Table 5 use α_f values as small as 12, for which S_α is not close to the Heaviside step in the usual uniform sense.
  3. [§5.3, Table 1] The complexity table is internally inconsistent with the algorithm. Table 1 reports training time O(LK) and memory O(LT)+λO(LKT), but Algorithm 1 constructs and stores the T×T matrix Λ, which alone is O(LT^2) memory if materialized per layer. It is also unclear what the coefficient λ denotes in a complexity expression. If Λ is never materialized and the matrix-vector products are applied implicitly via a scan or a recurrence, this should be stated explicitly; as written, the memory comparison undercounts the cost of the method and makes Table 1 difficult to interpret against the BPTT entries.
minor comments (5)
  1. [Table 2] In the ImageNet-100 block, the method name 'Normal BPPT' appears to be a typo for 'BPTT'.
  2. [Table 5 and surrounding text] The caption of Table 5 says the hyperparameters correspond to 'Table 1 in the main text,' but the relevant experimental results are in Table 2; Table 1 is the complexity comparison.
  3. [Table 5] The hyperparameter λTET is used in Table 5 but is never defined or explained in the text.
  4. [§2.2 and Table 4] The abbreviation IPSU is used in Section 2.2 and Table 4 but is never expanded or defined, and the related term RPSU is mentioned without a definition or reference.
  5. [Figure 2(c)] The caption says FPT achieves 'linear speedup' as T increases, but the axes and the reference baseline for the speedup factor are not specified, making the claim hard to verify from the figure alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FPT fixed-point scheme is an algebraic reformulation of the LIF recurrence, the PSN/PSU special-case claims are comparisons with independent prior models, and the evaluation is externally benchmarked.

full rationale

The derivation chain is self-contained. Equation (9) is obtained by unrolling the LIF recurrence Eq. (1) into vector form with the Toeplitz decay matrix Λ from Eq. (8), so the fixed-point map Φ(u) in Eq. (10) is equivalent to the original dynamics by algebraic rearrangement rather than by assuming the desired conclusion. The convergence analysis in Lemma 5.1 and Appendix A relies on the Banach fixed-point theorem and Hölder's inequality, both external mathematical facts, and computes its contraction constant from Λ and the surrogate Lipschitz constant without presupposing convergence of FPT. The relation to PSN and PSU in Section 5.2 is a genuine comparison with externally defined prior models: PSN's update u=Ac is shown to coincide with the K=1 iteration of the generalized Eq. (22), and PSU with K=2, so neither model is defined in terms of FPT. Experimental accuracy claims are tested against independent baselines (LocalZO, BPTT, OTTT, SLTT, etc.) on standard datasets, making them externally falsifiable. There is no load-bearing self-citation and no imported uniqueness theorem from the authors' own prior work. The concern that the O(K) time-complexity claim omits the cost of the T-dimensional Λ operations is a substantive correctness or complexity-modeling critique, but it is not a circularity: the headlined claim does not reduce by construction to a fit, a definition, or a self-referential citation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a small number of hand-chosen parameters (K, alpha_f, alpha_b) and on the unproven approximation of hard-threshold LIF by the sigmoid fixed point. The most consequential assumption is the cost model for Lambda operations, which is not stated or justified.

free parameters (3)
  • K (fixed-point iteration count) = 3 (typically)
    Chosen from the empirical observation in Fig. 2(a) that K>=3 yields negligible changes; Table 9 shows K=3,4,5 give identical accuracy on DVS-Gesture. The value is central to the claimed O(K) complexity.
  • Forward surrogate steepness alpha_f schedule = [3,12,12] on DVS-CIFAR10 and DVS-Gesture; [2,6,12] on ImageNet-100; [1,3,12] on sequential CIFAR
    A per-dataset hand-designed schedule that begins small for fast convergence and increases to approximate the step function; the specific values affect both convergence and approximation quality and are tuned per benchmark.
  • Backward surrogate steepness alpha_b = alpha_f / 3
    Set heuristically as one third of the forward value to smooth gradients; no theoretical justification is given, and it influences training stability and final accuracy.
assumptions (4)
  • domain assumption The LIF neuron update (Eq. 1-2) is the correct model for the tasks under study.
    All experiments use this neuron; the method's goal is to simulate this specific recurrence.
  • ad hoc to paper Replacing the Heaviside step with sigmoid S_alpha in the iteration, and then taking hard spikes at the end, yields a close approximation to the original LIF dynamics.
    No bound is given for the difference between the surrogate fixed point and the solution of the hard-threshold system; the paper relies on the empirical convergence plots in Fig. 2 (Section 6.1).
  • standard math Banach fixed-point theorem applies to the surrogate mapping in Eq. (11).
    Invoked in Lemma 5.1 and Lemma A.1; the theorem itself is standard, though the contraction constant is computed with a norm error.
  • ad hoc to paper The matrix-vector products with Lambda and Lambda-I have negligible or constant cost with respect to T.
    Section 5.3 and Algorithm 1 assume the per-iteration update is O(1) for the time-complexity claim; the T x T matrix structure and its computational cost are not addressed.

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Pith. "Pith review of Efficient Parallel Training Methods for Spiking Neural Networks with Constant Time Complexity." pith.science (2026). https://pith.science/paper/K2W32HTS

@misc{pith2026250612087,
  author       = {Pith},
  title        = {Pith review of: Efficient Parallel Training Methods for Spiking Neural Networks with Constant Time Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2W32HTS}},
  note         = {Machine review of arXiv:2506.12087}
}
abstract

Spiking Neural Networks (SNNs) often suffer from high time complexity $O(T)$ due to the sequential processing of $T$ spikes, making training computationally expensive. In this paper, we propose a novel Fixed-point Parallel Training (FPT) method to accelerate SNN training without modifying the network architecture or introducing additional assumptions. FPT reduces the time complexity to $O(K)$, where $K$ is a small constant (usually $K=3$), by using a fixed-point iteration form of Leaky Integrate-and-Fire (LIF) neurons for all $T$ timesteps. We provide a theoretical convergence analysis of FPT and demonstrate that existing parallel spiking neurons can be viewed as special cases of our proposed method. Experimental results show that FPT effectively simulates the dynamics of original LIF neurons, significantly reducing computational time without sacrificing accuracy. This makes FPT a scalable and efficient solution for real-world applications, particularly for long-term tasks. Our code will be released at \href{https://github.com/WanjinVon/FPT}{\texttt{https://github.com/WanjinVon/FPT}}.

Figures

Figures reproduced from arXiv: 2506.12087 by the authors.

Figure 1
Figure 1. Comparison of the forward and backward procedures of BPTT and FPT. In BPTT, the neuron processes sequentially over T timesteps, computing step-by-step through time. In contrast, FPT processes all timesteps simultaneously in K iterations, where K ≪ T. Circles represent scalars, and rounded rectangles represent vectors. membrane potential exceeds Vth, the neuron fires a spike: st = H(ut − Vth), (2) where H(·) is the H… view at source ↗
Figure 2
Figure 2. Convergence and efficiency of the LIF neuron using our proposed FPT method. The absolute error in membrane potential, spike firing error rate, and speedup factor relative to the original LIF are reported. Solid lines show the mean values across three experiments, with shaded areas representing the standard deviation. compromising accuracy? Finally, is it necessary to preserve the complete neural dynamics of LIF neur… view at source ↗
Figure 3
Figure 3. Comparison of accuracy and speed between FPT and the BPTT-like algorithm (LocalZO). Time indicates the average training time per batch on a single RTX 3090 GPU, with both models using the same architecture and hyperparameters. the static ImageNet-100 dataset, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Learned decay matrices for different parallel spiking neurons [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

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