Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This paper extends prior work [3,4] termed Implicit Differentiation on the Equilibrium state (IDE) to spiking neural networks. The authors derive a learning rule for the weights that is a product of two local equilibrium points in 2 phases. They perform experiments to compare their method with other SOTA SNNs, outperforming some.
main_review: The authors work is both an extension of previous work [3,4] and a generalization of equilibrium propagation [34]. The key contribution is the extension to spiking neural networks and finding a seemingly local learning rule for the weights. Of course, key issues in biological plausibility remain, which the authors briefly allude to, namely requiring two phases to compute the equilibria $a^*$ and $\beta^*$, how the masking would take place, how $\frac{\partial \mathcal{L}}{\partial a^*}$ would be computed and fed in, etc. These are an intriguing possibilities that require further work. Their network + training outperforms SOTA SNNs in some benchmarks with less neurons / params and time steps. While this work is incremental, the paper could be an important step in improving and incorporating on-chip neuromorphic learning, given the emerging importance of energy-efficient neuromorphic computing and the possibilities this paper raises for online local learning. While the mathematical extensions to spiking neural networks are worked out, these latter aspects of local / bio-plausible / neuromorphic learning are not fleshed out, which greatly limits the applicability of the work. Overall, the work itself is clearly presented with some promising aspects, though raising further questions as above.
limitations_and_societal_impact: Yes
review_title: null

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: The authors propose a relatively novel way of training spiking networks by considering the equilibrium state of their firing rates in response to input, and training using implicit differentiation, as has recently been explored in deep equilibrium models and implicit layers. They consider leaky and perfect integrate-and-fire neurons, as well as single layers with recurrent feedback, and hierarchical layers with feedback from the last layer to the first. The authors prove that the dynamics do indeed converge to an equilibrium state, and demonstrate the performance of the model on several state-of-the-art datasets.
main_review: The method of using implicit differentiation to train spiking networks appears novel and is very interesting! The description of the model dynamics and implicit differentiation is very clear (for the most part), and the results show that this method is indeed powerful and promising.

However, the main criticism of this work that I have is the lack of illustration of the model details. The paper jumps straight to the results, so while the authors show that the model works, no insights are gained beyond that. To aid the reader in understanding, it would be useful to have a couple of schematics illustrating the spiking network architecture for the single layer and hierarchical case. It would also be useful to show an example of the network responding to input over time, and comparing the spikes themselves with the estimates of the average firing rates in equations 6 and 9. This is a crucial part of getting implicit differentiation to work, so it seems important to show an example or two. Also, when thinking about biological plausibility and applications in neuromorphic hardware, the number of spikes fired is important, and it would be interesting if the authors can show some statistics on the average firing rate per neuron for the trained models.

Next, I have two more curiosities that I think are worth elaborating on in the paper a bit. One is about the stability of the equilibrium state during learning. The authors show that given particular conditions, the network will converge to equilibrium, but what’s stopping the network from violating these conditions during training? E.g., does the network ever evolve into limit cycle behavior? Second, while the authors show convergence in Fig 1, I am left wondering how good this convergence needs to be in order to have effective learning. It is perhaps too much for an exhaustive study of results with different number of time steps, but perhaps the authors could discuss where both classification and learning might break down due to insufficient time steps simulated and thus accumulated errors.

Lastly, there are a few related works that are worth citing, including Li & Pehlevan 2020 and Mancoo et al. 2020, both of which appeared in neurips last year. These papers describe spiking network models that do evolve to equilibrium states, defined by an implicit function (in those cases the implicit function happens to be a constrained optimization problem). They do not scale them up to deep learning problems, but they have already explored spiking networks evolving to equilibrium states.

limitations_and_societal_impact: The authors did not include a broader impact statement.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a new training method for SNNs that relies on decoupling the forward propagation from the backward by viewing the SNNs as solving a fixed-point equilibrium equation and exploiting implicit differentiation results to compute the gradients. The theoretical contribution is to derive the equilibrium states for the fixed-point equations that the SNNs are solving, based on which the implicit differentiation technique can be leveraged. The practical contribution is a training algorithm that avoids the problems of non-differentiability in SNNs and achieves state-of-the-art performance on many datasets. 

main_review: The paper proposes a new training method for SNNs that relies on decoupling the forward propagation from the backward by viewing the SNNs as solving a fixed-point equilibrium equation and exploiting implicit differentiation results to compute the gradients. The theoretical contribution is to derive the equilibrium states for the fixed-point equations that the SNNs are solving, based on which the implicit differentiation technique can be leveraged. The practical contribution is a training algorithm that avoids the problems of non-differentiability in SNNs and achieves state-of-the-art performance on many datasets. 
 
Highlights:
 
1. The most important contribution definitely is a training algorithm that does not rely on the exact forward procedure. The SNN literature is full of studies trying to overcome the issue of non-differentiability through pseudo-grad, or large memory requirements using simpler spiking models. There are very few studies that avoid these problems altogether and this is one of them. 

2. The authors provide rigorous proof of the equilibrium states that the SNNs converge to under both continuous and discrete dynamics. As a result, one can leverage implicit differentiation models to compute the gradients w.r.t. the fixed point equations, thus avoiding the need for a backward procedure that relies on the forward procedure. 

3. Based on 1, the authors propose a training algorithm for SNNs with the assumption that the average firing rates of the SNNs follow the fixed-point equations. The loss and gradients of the loss become straightforward to compute. 
While not completely obvious at first, the authors provide a qualitative study of how the learning rule is similar to the Hebbian learning rule, thereby imparting it some degree of biological plausibility. 

4. Experiments on many datasets, including both small benchmark datasets and large image datasets, show that the algorithm can obtain state-of-the-art performance with a much less number of network parameters and timesteps. This demonstrates the practical effectiveness of the method.
 


Concerns:

1. From equation 10, it doesn’t seem like the network is weight-tied and does not have input injection at each layer like in [1]. As a result, acc. to Theorem 3, the equilibrium point does not directly depend upon the inputs which might affect its stability. That’s not the case in [1] from which the method seems to have been inspired. Can the authors resolve this discrepancy, and conduct additional experiments on weight-tied deep networks?

2. How practical are the assumptions in Theorem 1? In particular, how does the fact that the membrane potentials and network weights need to be bounded affect the initialization of these hyperparameters? This is hinted at in the appendix, but I think it deserves a discussion in the main text. Moreover, why is alpha in equation 25 in appendix D a learnable parameter? 

3. Is it fair to characterize the learning algorithm as rate-based? If that is the case, and combined with the fact that the inputs need to be convergent, how does the algorithm exploit the temporal capabilities of the SNNs? Is an event-based version of this algorithm possible?
 


Minor Comments:

1. In the introduction, the authors mention that “SNNs naturally compute with multiple time steps, which inherently supports feedback computation”. Yet, authors include a feedback connection in their architecture. Can the authors discuss what is the difference in the feedback information represented by these two mechanisms? In addition, it would be nice to have an ablation study of the network without feedback connections. 

2. It seems like one advantage of the method is the decrease in training complexity over other ways of training the networks (such as STBP [2]). This might be due to the fact that backward computation relies on a simpler procedure than following the exact forward propagation. If that is the case, can the authors quantify the ease in training and include a short discussion. 

3. It will be more clear if the authors can include algorithm pseudocode for the training algorithm.

[1] Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models. In Advances in Neural Information Processing Systems, volume 32, pages 690–701, 2019.

[2] Yujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. Spatio-temporal backpropagation for training high-performance spiking neural networks. Frontiers in Neuroscience, 12:331, 2018.

limitations_and_societal_impact: The proposed training algorithm for SNN relies on equilibrium states on firing rates for each layer of the network. This limits the SNN for non-stationary inputs (time-series data that changes over time, for example, audio and video) where such equilibrium states may not exist in the network.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper presented a training method for feedback spiking neural networks to tackle existing training problems with backpropagation framework. Experiments were conducted on five image datasets, MNIST, Fashion-MNIST, N-MNIST, CIFAR-10 and CIFAR-100, superior performances were claimed.  
main_review: The idea of this paper is novel, training a robust deep SNN is very critical in brain-inspired learning field. From the prespective of equilibrium of NNs is impressive, however, I didn't see more details about the equilibrium of SNN. Based on BP, the authors proposed IDE method to try to train a deep SNN. Experimental results evaluated on MNIST and CIFAR showed that the proposed model could achieve good classification performance on those datasets with fewer time steps.

However, there are some missing descriptions. What is the difference when you applied IF or LIF, and which situation did you prefer IF of LIF? In section 3.2, when you choose the fixed-point operation, how can you guarantee the convergence？

The experiments conducted on MNIST or CIFAR based were not enough when you proposed a BP-based model to train a deep SNN. I prefer to see the results from ImageNet datasets or analysis of energy consumption.


limitations_and_societal_impact: The writing is clear and the motivation is clarified clearly. Besides, the theoretical grounding and experimental evaluation are not sufficient to show their originality and significance. Here are some of the suggestions: 
1)	I would like to see ablation studies for the proposed training method, the traditional backpropagation framework refers as the baseline. 
2)	How to deal with the different types of inputs (e.g., bio-medical signals or speech)? It would be valuable to discuss it and present your solutions in this paper. 
The citation seems a bit disordered.
review_title: null

Meta-review: The paper proposes a novel method to train spiking neural networks (SNNs) by considering the equilibrium state of their firing rates. They exploit results on implicit differentiation to compute gradients in the network. They show that the method achieves state-of-the-art performance on several data sets, including CIFAR-10 and CIFAR-100.
The method is based on rigorous proofs about the equilibrium states of an SNN.

All reviewers agree that the paper presents interesting results. The manuscript is well-written and the experimental results are convincing.
Decision: Accept (Spotlight)