Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: By restricting (can also be viewed as expanding) to holomorphic neural networks, the authors are able to use a contour integral in the complex plane for finite $\beta$ to evaluate a differential used in equilibrium propagation (EP) that is only valid as an approximation of the backprop gradient in the limit $\beta \to 0$. The authors interpret this use of the contour integral as a time integral of an oscillating teaching signal. The authors show that this holomorphic EP surpasses EP in performance especially in the presence of noise.

strengths_and_weaknesses: The authors incorporate a cool idea to convert a differential to a contour integral in the complex plane and then interpret this integral as a time integral of an oscillating teaching signal. They also show that allowing for finite beta enables robustness to noise, unlike EP which is only vaild in the lim $\beta \to 0$. However, the authors do not provide any pointers as to how this may be implemented in biology except mentioning that there may be connections to theta neurons and phasor networks. Still this is an interesting idea that possibly deserves to be disseminated to stimulate further progress in the search for bio-plausible back-prop.
Disclaimer: I have not verified any proofs / derivations.
questions: Perhaps the authors can speculate further in the discussion how such a scheme may be implemented bio-plausibly.

[Update: since the authors have included this, I have enhanced my rating from 7 to 8.]

Minor corrections:
Figure 2 legend for panel d "circular path in panel d)" -> "circular path in panel b".
inconsistency in lines 85-86: $l(y,\theta,s)$ on line 85 vs $l(\theta,s,y)$ on line 86, also line 91.

limitations: Yes
review_title: null

final_score: rating: 9: Very Strong Accept: Technically flawless paper with groundbreaking impact on at least one area of AI/ML and excellent impact on multiple areas of AI/ML, with flawless evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: Equilibrium propagation is a promising framework to learn neural networks in a biologically plausible way. However, it is sensitive to noise, which makes it hard to scale to large machine learning tasks, and requires two phases, limiting its interpretability as a biologically plausible learning framework. The authors solve those two problems by introducing a new variant of equilibrium propagation that relies on complex analysis and holomorphic functions, and derive an online unbiased version of the corresponding learning rule. They show that this new algorithm is more robust to noise than traditional equilibrium propagation and scales to large vision classification tasks such as ImageNet.
strengths_and_weaknesses: + Significant theoretical advances on equilibrium propagation
+ Careful empirical study of the impact of the different parameters ($N$, $\beta$) on the quality of the gradient approximation
+ Strong empirical results (robustness to noise, scale to ImageNet) while being arguably more plausible than previous equilibrium propagation versions
+ The paper is overall very well written
-  (Weakness) A detailed description of the algorithm is lacking, along with an interpretation in terms of neural dynamics
questions: The suggestions below aim at improving the main weakness of the paper:

- It is hard to understand how $s_\beta^*$ is computed when $\beta$ is a complex variable. A naive interpretation would be that both the real and imaginary parts of $\partial_s F$ have to be 0. However, this does not require $s$ to be a complex variable, which seems to be the case in the paper (cf. Eq. 3 or Fig.2.a). More details on how $F$ is defined, how $s^*_\beta$ is computed, and which signal the real and imaginary parts of $s$ encode would be very valuable.

- In the paper and in the appendix, the authors assume that $\partial_s F$ is a holomorphic function. How big of an assumption is this? Do we have it for free by construction or are there some additional assumptions compared to the real version of equilibrium propagation?

- Throughout the paper, a basic understanding of complex analysis is assumed. A short background section in the appendix containing the tools (holomorphic function, Cauchy-Riemann equation, holomorphic IFT...) needed to appreciate the theoretical contributions of the paper would be useful for the reader unfamiliar with the topic.

Minor points:

Section 3 - Theoretical results
- line 122: "The proof is an extension of the one provided for the real nudging case [10]"

- In Equation 3, are the * superscript missing on $s_\beta$? the notation $s_\beta(\theta, s_\beta, \beta)$ is quite misleading, can you clarify it?

- For the Cauchy integral of equation 5 you need the implicit function $s_\beta^*$ to be defined on the path $\gamma$, which is not in the neighborhood of 0. So Theorem 1 should be more general and ensure that the IFT conditions are defined on the entire path. Can you make this point clearer, either in the main text or in the appendix?

Section 4 - Experiments
- In Table 1, the comparison between classic EP and holo EP is not fair in the noiseless case: cEP uses 200 + 1x100 steps per gradient when hEP uses 200 + 10x100. Increasing the total number of steps of cEP to 1200 (and decreasing $\beta$ to benefit from this extra number of steps) would make the comparison fairer.

- In Table 1, the cEP estimate used in the noisy regime uses 1 point when $\beta$ is zero and 10 for non-zero $\beta$. This results in a very biased negative update but unbiased positive update. One other strategy would be to use 5 points for the two phases to reduce this "bias imbalance". I would suspect this to improve the quality of the cEP estimate in the noisy regime.

- Figure 3.a) and 4.a) plot "cosine dissimilarity" when Figure 2.d) and 3.c) use "cosine similarity". Using the same measure would make navigation in the figures easier.

- Legend Figure 3.a), are $T_\text{dyn}$ and $T_\text{plas}$ fixed? If so, to which values?

- Line 236-238: refer to Zucchet et al. 2021, which theoretically analyzed this behavior?

- It is unclear to me how important is the choice of the dSiLU activation function. On one side, it seems necessary theoretically to have holomorphic dynamics. On the other, it seems to allow for a more flexible choice of $\beta$ (see Fig. 7), which could suggest that it makes convergence to equilibrium faster. I’m therefore wondering how crucial is this activation for the good empirical results displayed in the paper.

Appendix
- Equation (12): with the current brackets it is not directly that the product is 0 because $\partial_s F$ and $\partial_{\bar{s}} F$ are. Moving them under the partials would make the proof easier to read.
limitations: I found the discussion of the limitations of the paper adequate.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors present `holomorphic equilibrium propagation' (holomorphic EP), an extension of equilibrium propagation for networks with holomorphic activation functions, which estimates the gradient of the loss using an approximation of the complex Cauchy integral. This avoids the need for an infinitesimal teaching signal, making the method more robust to noise.
strengths_and_weaknesses: Strengths

I think the authors have presented a beautiful extension of equilibrium propagation for energy-based models with holomorphic activation functions. I found their presentation to be clear and as far as I can tell, their work is original (though I am not an expert in this area). The problem of solving the credit assignment problem using local learning rules is an important challenge in neuroscience and I believe the authors have put forth an interesting and elegant solution.

Weaknesses

Despite the elegance of the authors' solution, it is not clear if the results here are actually useful for understanding neural computation, which is in part the motivation for studying local learning rules. However, I think it is reasonable to leave such considerations for future work.
questions: For someone who is not an expert, Figure 2 was challenging for me to interpret. Perhaps the figures can be connected more directly to the equations?

Minor questions/comments:

Line 143: it's not clear to me why $\text{d}\beta=(2\text{i}\pi\beta/T)\text{d}t$. Is there a $e^{2\text{i}\pi t/T}$ term missing?

Line 210: The NeurIPS 2020 paper "A biologically plausible neural network for slow feature analysis" may be relevant here.

Line 216: should "discreetly" be "discretely"?
limitations: As stated above, it is not clear to what extend the method of holomorphic equilibrium propagation is useful for understanding neural computation. The authors address this to some degree in the discussion, but not in detail. I think this is fine for this paper, but it is an important potential limitation.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors begin by showing that the classic equilibrium propagation can be extended to the case where the network layers are holomorphic dynamical systems. The essential result which follows is that the gradient with respect to parameters can be transformed to a contour integral on a circle around zero, which is equivalent to the first Fourier coefficient of the derivative of the nudged energy function with respect to network parameters. The central claim is that this equivalence eliminates the requirement that the "nudge size" go to zero. The authors define a straightforward numerical approximation to the gradients (equivalent to discrete Fourier transform), and show that the bias of this approximation goes to 0 as the sample rate of the Fourier transform goes to infinity.

The authors' toy experiment shows that holomorphic EP is capable of learning and that the cosine similarity of the approximated gradients is significantly higher than with classic EP.

The authors' theoretical investigations also reveal that holomorphic EP can be adapted to an online learning algorithm, assuming that the time scale of the layer dynamics is much less than the time scale of the oscillations computing the Fourier transform, which is in turn much less than the time scale of the weight updates. The authors validate this result with the MNIST task and show that the online approximation of the gradients are still significantly more accurate than the classic EP algorithm. They also compare classic, homlomorphic, and online homlomorphic EP on the MNIST task and claim that holomorphic EP has greater robustness to noise.

Finally, the authors validate their algorithm by adapting a CNN architecture to be holomorphic and showing that the holomorphic EP training yields virtually the same results as BP training on a standard CNN.
strengths_and_weaknesses: The paper presents a very novel idea which greatly increases the power of a local learning rule using the elegance of complex analysis.

The proof of theorem 1 in the main text could be presented more clearly. It seemed as though the authors were implying that the logic of the complex chain rule needed to be applied to $\theta$ as well as $\beta$. The proof in the appendix is much more clear, I think that one thing the authors could do to improve presentation in the main text would be to denote $s_{\theta,\beta}$ instead of $s_{\beta}$.

The derivation of the weight gradients presented was only with respect to a mean-squared error loss. However, for the computer vision tasks I would assume cross entropy loss would be required. Was an equivalent version of Eq. 8 derived for cross entropy anywhere? Eq. 8 seemed out of place since the main tasks are classification. If I understood correctly, Eq. 8 is the basis for the derivation of the online learning algorithm, is the use of MSE in the online learning essential or inconsequential? It would be strange if the online training of MNIST used MSE. Was the JAX autodiff engine used to compute the gradient of the nudged energy function with respect to parameters, in the case where MSE was not used?

The toy task was not adequately explained in the main text, what exactly are the inputs and targets? The way it sounded was that the network is given noise as input and is supposed to output a single value. This is very trivial, even for a toy task. Were there multiple Gaussians? Was the task to classify which samples came from which distribution? How many classes were there?

Another minor clarity issue is the labelling of Figures 2d and 3c versus 4a and S6. Why did the authors choose to present 1 - cos similarity instead of just the similarity itself? This made the plots look very surprising at first.

Overall, this is a very interesting paper which I would very strongly accept on the condition that its clarity issues are fixed.
questions: A major numerical drawback of holomorphic functions is the presence of poles and unbounded behavior. Both the sigmoid and dSilU activation functions will have poles where ever the exponential function takes the value -1. Why do these poles not cause any kind of instability in the numerical implementation?

Since the learning algorithm can be adapted to be online, it was a bit of a disappointment to not see any time-domain tasks. The authors mention this as a limitation in their discussion. However, why would the algorithm still not work if the time scale of the task was much slower than the time scale $T_{dyn}$ of the layer dynamics? Was this ever tested?


limitations: The authors adequately addressed potential societal impact and some of the limitations of their work. However, only in the final appendix did I find a note that holomorphic EP was *much* slower at learning than BP. Especially since the authors mention edge and IoT devices in their discussion, I think a discussion of computational cost should also be moved to the discussion. Just how much longer should we expect holomorphic EP to take to train and evaluate? Was it only for ImageNet, and why would that be the case? It was not made clear whether or not 64 bits was required for each real and imaginary part or for the entire complex number, this is an important factor which would certainly affect performance.
review_title: null

Meta-review: metareview: Equilibrium propagation is a biologically plausible form of backpropagation based learning where the true gradient of an energy based model is computed for infinitesimal perturbations. In this innovative work, authors extend EP using complex analysis that links contour integrals for finite perturbations with the oscillatory dynamics in time. This not only allows better gradient estimates, but also applications to related theories of learning in neuroscience as well as neuromorphic engineering. It represents a significant advance that opens new doors.

confidence: Certain
Decision: Accept