Improving equilibrium propagation without weight symmetry through Jacobian homeostasis
Authors: Axel Laborieux
Friedemann Zenke
Venue: ICLR 2024 poster (2024)
Keywords: Equilibrium propagation
local learning
weight transport
jacobian regularization
Primary area: learning theory
TL;DR: We disentangle the impacts of weight asymmetry and finite nudge on the learning performance generalized equilibrium propagation, and propose a new homeostatic objective to recover good performance.

Abstract: Equilibrium propagation (EP) is a compelling alternative to the back propagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. 
Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to yield unbiased gradient estimates.
Both requirements are challenging to implement in physical systems.
Yet, whether and how weight asymmetry contributes to bias is unknown because, in practice, its contribution may be masked by a finite nudge. 
To address this question, we study generalized EP, which can be formulated without weight symmetry, and analytically isolate the two sources of bias.
For complex-differentiable non-symmetric networks, we show that bias due to finite nudge can be avoided by estimating exact derivatives via a Cauchy integral.
In contrast, weight asymmetry induces residual bias  through poor alignment of EP's neuronal error vectors compared to BP resulting in low task performance.
To mitigate the latter issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's fixed point. 
This homeostatic objective dramatically improves the network's ability to solve complex tasks such as ImageNet 32$\times$32. 
Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's relaxation dynamics.

Published as a conference paper at ICLR 2024
IMPROVING EQUILIBRIUM PROPAGATION WITHOUT WEIGHT SYMMETRY THROUGH JACOBIAN HOMEOSTASIS
Axel Laborieux', Friedemann Z Friedrich Miescher Institute for Biomedical Research, Basel, Switzerland Faaultyy of Science, University of Basel, Switzerland (f fiirmtname. lastname) ) fmi.ch
ABSTRACT
Equilibrium propagation (EP) compelling alternative the backpropagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to yield unbiased gradient esti- mates. Both requirements are challenging to implement in physical systems. Yet, whether and how weight asymmetry contributes to bias is unknown because, in prac- tice, its contribution may be masked by a finite nudge. To address this question, we study generalized EP, which can be formulated without weight symmetry, and ana- lytically isolate the two sources of bias. For networks, we show that bias due finite nudge can be avoided by estimating exact derivatives via a Cauchy integral. In contrast, weight asymmetry induces residual bias through poor alignment of HP's neuronal error vectors compared BP resulting in low task performance. To mitigate the latter issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's 's fixed point. This homeostatic objective dramatically improves the network's s ability to solve complex tasks such ImageNet 32x32. Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's 's relaxation dynamics.
non-symmetric
INTRODUCTION
Virtually all state-of-the-art artificial intelligence (AI) models are trained using the backpropagation of error algorithm (BP) (Rumelhart et al., 1986). BP, neural activations are evaluated during the forward pass, and gradients are obtained through corresponding backward pass. While BP can be implemented efficiently on digital hardware which simulates neural networks, it is less suitable for physical neural networks such as brains neuromorphic substrates, which operate lower energy costs. This limitation is mainly due to two technical requirements the backward pass in BP. First, the backward pass linear while leaving neuronal activity unaffected (Lillicrap et al., 2020; Whittington & Bogacz, 2019). Second, back-propagation requires the transpose of connectivity matrices, often referred to "weight symmetry" (Grossberg, 1987). For these reasons, many alternative learning algorithms that can run on dedicated neuromorphic hardware have been designed (Boybat al., 2018; Thakur et al., 2018; Indiveri al., 2011; Schuman al., 2017: Frenkel & Indiveri, 2022; Yi et al., 2023).
One such alternative learning algorithm is equilibrium propagation (EP)(Seellier Bengio, 2017), which can provably estimate gradients using only the network's 's own dynamics. Classic EP requires an energy-based model (EBM), such as differentiable Hopfield network (Hopfield, 1984), which relaxes to an equilibrium for any given input. EP computes gradients by comparing neuronal activity at the free equilibrium to the activity at second equilibrium that is "nudged" toward a desired target output. Preliminary hardware realization dedicated EP suggests that it could reduce the energy cost of Al training by four orders of magnitude (Yi et al., 2023). Unlike BP in feed-forward networks, HP uses the same network propagate feed-forward and feed-back activity, thereby dispensing with

---

Published as a conference paper at ICLR 2024
the need linearly propagate enror vectors. In particular, EP is promising candidate for learning through neuronal oscillations (Baldi & Pineda, 1991; Laborieux & Zenke, 2022; Anisetti et al., 2022; Delacour et al., 2021), which is why we focus on EP in this work.
Nevertheless, EP requires weight symmetry and vanishing nudge for unbiased gradient estimates, which limits its potential for neuromorphies compared other better performing BP alternatives (Ren et al., 2023; Journé et al., 2023; Payeur et al., 2021; Greedy et al., 2022: Høier et al., 2023). While Laborieux & Zenke (2022) showed that an oscillation-based extension of EP called holomorphic EP (hEP) removes the bias by integrating oscillations of neuronal activity, this approach has only been demonstrated using symmetric weights. Although generalization of EP to non-symmetric dynamical systems exists, the approach has only been demonstrated on simple tasks like MNIST (Scellier al., 2018; Ernoult al., 2020; Kohan al., 2018), while it fails train on CIFAR-10 10 (Laborieux et al., 2021). Previous work investigated the effect of non-symmetric feedhack on training with BP (Lillicrap et al., 2016; Nøkland, 2016; Launay et al., 2020; Clark al., 2021; Refinetti et al., 2021) and approximate BP (Hinton, 2022; Dellaferrera & Kreiman, 2022; Löwe et al., 2019), but little attention has been given to effect of weight symmetry for EP. this article we fill this gap. Overall, our main contributions are:
We provide comprehensive analysis of the individual sources of bias in the gradient estimate from weight asymmetry and finite-size nudge in generalized EP.
An extension of hEP (Laborieux & Zenke, 2022) to non symmetrie complex-differetiian dynamical systems, that can be estimated through continuous oscillations.
We propose a new homeostatic loss that reduces the asymmetry of the Jacobian at the free equilibrium point without enforcing perfect weight symmetry.
An empirical demonstration that hEP with homeostatic loss scales to ImageNet 32x with a small performance gap compared to the symmetric case.
2 BACKGROUND
2.1 GRADIENTS IN CONVERGING DYNAMICAL SYSTEMS
Let us first recall how gradients are computed in converging dynamical systems (Baldi, 1995). Let F be differentiable dynamical system with state variable u and parameters O. A subset of units receive static input currents æ. The dynamics of the system are governed by:
du dt
F(0,u,))
We further assume that the dynamics of Eq. (1) converge a fixed point of the state u. This phase of the dynamics is defined as the "free phase", because there are no contributions from any target y associated to æ. While stable fixed points provably exists for EBMs (Hopfield, 1984), there is no guarantee to converge for arbitrary dynamical systems. However, a fixed point exists if F is contraction map (Scarselli al.. 2008; Liao al., 2018). In the following, we just assume the existence of fixed point and denote it by un:
0 F F(0, ug x).
In ugj the superscript x indicates convergence to a fixed point, and the subscript 0 stands for the "free" phase. We further assume that ui is implicit function of O, such that its derivative exist by means of the implicit function theorem (IFT). Given an objective function L that measures the proximity of subset of u* (output units) to a target y, the gradient of L with respect to 0 is obtained by the d.C. duy chain rule through the fixed point ae aug de where dc, is the error at the output units, and du; de
aug
is obtained by differentiating Eq. (2) with respect to 0:
OF
du
0
(uj) + JF JF(
00
d0
du
( -1 OF
00 (4))
(3)
d0
2

---

Published as a conference paper at ICLR 2024
a)
Energy-based
b) Non energy-based
c) Feed forward
Layer
OOOO
COOO W 1 ji OOOAO 1 1
p000 Wij
Layer
O
OOOOO
Layer
C
OOCOO
COOO
Ø Lower-triangular
Symmetric Jacobian
Non-symmetric
Lapers Layer Layes
Figure Different neural network architectures and their Jacobian after the free phase. a) A continuous layered Hopfield network. The existence of energy function enforces Jacobian symmetry with reciprocal and tied connectivity Wij Wji- b) A layered network with reciprocal connectivity but independent forward and backward connections, which make the Jacobian symmetric. e) A discrete feed forward network without feedback connections. The Jacobian is lower triangular, allowing the explicit back propagation of error layer by layer.
Du Where J=(()) == OF (0, up: x)) is the Jacobian of the network at the fixed point (Fig. 1). Replacing
dug in the expression the gradient de and transposing whole expression yields (Baldi, 1995); de
dc d0
OF 00 (u))
-Jp((n)) -T dc
=
du
"pre-synaptic"
"post-synaptic"
We refer to the part of the expression in parenthesis as the neuronal error vector ò. the widely used reverse-mode automatic differentiation (AD), linear system involving the transpose of the Jacobian solved to compute 5 from the output error dug matrix involving pre-synaptic variables. In feed forward networks, the lower-triangular structure of Jacobian allows computing recursively layer by layer from the output (Fig. Ic) as exploited in BP (Rumelhart et al., 1986). In converging dynamical systems, recurrent backpropagation (RBP) and variants (Almeida, 1990; Pineda, 1987: Liao et al., 2018) obtain ö as the fixed point of an auxiliary linear dynamical system using Jr(u)) T_ In deep equilibrium models (Bai et al., 2019) 5 is found with a root-finding algorithm. In the next section, we review how ö is obtained in EP.
dc and finally multiply with 0F 80 (u6). which a sparse
2.2 NEURONAL ERRORS IN EQUILIBRIUM PROPAGATION
The EP (Scellier & Bengio, 2017) gradient formula for EBMs can be derived without explicit of the energy function formalism (Scellier et al., 2018), by noticing a similar pattern between the derivative of uj with respect to parameters 0 (Eq. (3)) and 8 in Eq. (4). While d0
duğ is obtained by
inversion of Jr (u()), 8 is obtained by inversion of Jr (u)) T. However, in EBMs for which there exists scalar energy function E such that F (Fig. la) due to the Schwarz theorem on symmetric second derivatives. The remaining difference between Eqs. (3) and (4) is the quantity which the inverted Jacobian is applied: 8F (u,) in Eq. (3)
dE have precisely Jr (uf) JJ (
and dc in Eq. (4). The solution is define parameter 33 such that these quantities are equal: dug
Definition 1 (Nudge parameter (3). Let 3 be scalar parameter that equal to zero during the free phase. justifying the notation uy and such that aF contributes to the dynamics through duy.
dc When B # O, the target de The term "teaching amplitude" refers to 1,31.
03 (u))
dug

---

Published as a conference paper at ICLR 2024
a)
Symmetric
C)
Second hidden layer
Membrane potentall 0 Membrone potential M*
Raw oscillation
1.00
10
Time average
Casloyer Spo
0.98
Decreasing
Free fixed point
symmetry
Density
0.94
0.1
0.92
0.0
0.90
b)
1.00
Non symmetric
d)
10
First hidden layer
0.4
Online first mode
CoalSge (pir))
0.98
0.3
Trua dju
0.96
Densty
0.2
0.94
0.92
2-
0.0
1.5
-1.0
-0.5
0.0
0.5
1.0
100
200
300
400
Teaching amplitude [F]
cost dg0))
Time
Figure 2: Separating the sources of bias due to finite nudge and Jacobian asymmetry. a,)) Cosine similarity between the neuronal error vector computed by RBP and classic EP (grey) (Scellier & Bengio, 2017; Scellier et al., 2018), holomorphic EP (purple and pink) for different number of points to estimate Eq. (7), in function of the teaching amplitude | for symmetric (a)) and non symmetric (b)) equilibrium Jacobian. c,d) Kernel density estimate of the cosine distribution between both neuronal error vectors in a two-hidden layer MLP. Alignment worsens with increasing asymmetry and depth. e) Continuous-ting time estimate of the free fixed point through oscillations induced by the teaching signal ((t). f) Continuous-ting time estimate of the neuronal error vector of generalized hEP.
By applying the same derivation as Eq. (3) for the new parameter we have:
du
= =-F((u)) 1 OF
dc
(u)) = = JF((()
= 8.
(5)
d3
03
du
8=
for EBMs
by Def.
Therefore, the EP neuronal error vector is the derivative of the fixed point un with respect to 3 in O, which denote by dyu" for short. Note that the derivation assumes that up is an implicit function of 3. In practice, dg is approximated by finite differences after letting the network settle second equilibrium dsuu (u² =5)/ 3, which contains bias due the finite 3 & 0 used in the second phase. The EP gradient estimate can then be written without using the energy function by replacing d by dyu" in Eq. (4):
OF
V
00 (u)) dguu.
(6)
Importantly, the quantities dg and V are defined for any differentiable dynamical system (Scellier et al., 2018), not just for EBMs. However, dzu" coincides with 5, and Vo with dc ae (Eq. (4)), only
for EBMs as their Jacobians Ju(u,) are symmetric. We provide more details on the relation between Eq. (6) and the usual gradient formula involving the energy function (Scellier & Bengio, 2017) in Appendix B. Intuitively, d can be seen as the replacement for 8 that the system can obtain using its own dynamics. When (r(u)) # JF(u))" (Fig. 1b), the asymmetric part of the Jacobian
dE contributes an additional bias to the estimate of de However, its contribution has never been studied independently because it is masked by the finite nudge 3 in practice. It is worth noting that d 0 for feed forward networks due to the absence of feedback connections (Fig. Ic).
THEORETICAL RESULTS
3.1 GENERALIZED HOLOMORPHIC EP
To study individual bias contributions build Holomorphic EP (Laborieux & Zenke, 2022). which, assuming weight symmetry, allows computing unbiased gradients with finite nudges. As we will see, the property generalizes to asymmetric weights and d can be exactly computed with
4

---

Published as a conference paper at ICLR 2024
finite teaching amplitude [ for any complex differentiable (holomorphic) vector field F. We use the fact that for holomorphic functions, derivatives at a given point are equal to the Cauchy integral over path around the point of interest B 0 (Appel. 2007) (see Appendix A.1 for more details):
du d3
1
%
d3
2itt/T dt.
T13 Here we used the teaching signal ((t) = |
2itr
y B2
8=0
E [0, T) for the path % and performed
the change of variable d, - ((1) 7 dl, where i E C is the imaginary unit. Moreover, Eq. (7) assumes that the fixed point u is implicit function of 3 and well-defined
the entire path y. In other words, we assume that the system is at equilibrium for all t E [0, T", which can be achieved when the timescale of the dynamical system is shorter than the one of the time-dependent nudge B(t). In numerical simulations, we estimate the integral with a discrete number of N points: //T A/N, k (0,-... N 1] (see Appendix E. Regardless of whether the dynamical system has symmetric Jacobian at the free fixed point or not (Fig. 1), the exact value of dy can be computed for finite range of teaching amplitudes |3] (Fig, 2a,b).
3.2 ESTIMATING GRADIENTS IN CONTINUOUS TIME
Although Eq. (7) implies that de is obtained at the end of one period of the teaching signal (t T), it can also estimated continuously by letting the integral run over multiple oscillation cycles of the teaching signal (()) (Fig. 2D:
1/8 //" ug(()e 2i0r// /T
1
du-
dr
(8)
This relation is made obvious by splitting the integral over all the completed periods, such that the relative contribution of the remainder integral asymptotically approaches 0 as + oo (see Appendix C.1). The advantage of this approach for physical systems is that dyu" need not be accessed at the precise times t = T, thereby dispensing with the need for separate phases (Williams et al., 2023). Moreover, the free fixed point does not need to be evaluated to obtain dyæu" However, the pre-synaptic term 30 (us) in Eq. (6) a priori still requires evaluating the free fixed point. We will see next that we can also estimate it continuously in time thanks to the Mean Value Theorem from complex analysis (Appel, 2007) (Fig. 2e), Appendix C.I):
OF / 00 (u (3) dd
0F
1-000 00 (4))-
Overall, the gradient estimate V in the time-continuous form corresponds to the average of pre- synaptic activation multiplied by the amplitude of the post-synaptic potential, consistent with models of STDP (Dan & Poo, 2004; Clopath & Gerstner, 2010). the next section, we isolate the contribution to the bias brought by the asymmetry of the Jacobian Jr(j)).
3.3 ISOLATING BIAS FROM JACOBIAN ASYMMETRY
We are now in the position to quantify the bias originating from asymmetric Jacobians in the computation of the neuronal error vector of hEP. By isolating OF and dL both ends of Eq. (5), du, and because they are equal by Definiton 1, we see that both the neuronal error vectors of hEP and RBP are linked by the following relation:
douu = J= (()
(10)
Thus, for asymmetric Jacobians, 5 and d not aligned which introduces bias in the resulting gradient estimates. To further quantify this bias, we introduce S and A the symmetric and skew- symmetric parts of Jr. such that Jr(uj) S A and J=(u)) when the Frobenius norm |||S-_ A|ll + 0. dyuu and 8 are linked by (see Appendix C.2 for proof):
S A. Then we can show that
døuu 8 - 2S A8 o A6).
(11)
This expression highlights the direct dependence of the bias on A and further suggests a way to improve the Jacobian's 's symmetry, thus reducing the bias by directly decreasing the norm of A.
5

---

Published as a conference paper at ICLR 2024
3.4 INCREASING FUNCTIONAL SYMMETRY THROUGH JACOBIAN HOMEOSTASIS.
To reduce the norm of A we note that || ||| - Ar = Tr(JF
Tr())), where the dependence of JF on u" is omitted. Similar to work by Bai et al. (2021) on deep equilibrium models, we can use the classical Hutchinson trace estimator (Hutchinson, 1989) (see Appendix A.2) to obtain the following objective for minimizing Jacobian asymmetry and, therefore, improving "functional symmetry":
Chomen
E E=N((.
[|]rE||2 s² JT]E]]
(12)
If we break down the two terms present in this objective, we see that ||| should be minimized, which corresponds to making the network more robust to random perturbations by minimizing the Frobenius norm of the Jacobian. Interestingly, Stock et al. (2022) showed how local heterosynaptic learning rule could optimize this objective locally through synaptic balancing (see Eqs. (12) (14) in Stock et al. (2022)), whereby neurons try to balance their net inputs with their output activity. The second term in Eq. (12) comes with minus sign and should therefore be maximized. is, thus, akin to a reconstruction loss, similar to the training of feedback weights in target propagation (TP) (Meulemans et al., 2020; Ernoult et al., 2022; Lee et al., 2015), which can also be local (see e.g. Eq. (13) of Meulemans et al. (2021)), and reminiscent of predictive slow feature learning (Wiskott & Sejnowski, 2002; Lipshutz et al., 2020; Halvagal & Zenke, 2022). However, in our case all weights are trained optimize Chomeo. unlike TP approaches which only train feedback weights. Overall, Lhomeo is a plausible objective for increasing and maintaining functional symmetry of the Jacobian in any dynamical system. Finally, it is important note that while perfect weight symmetry implies functional symmetry, the converse is not true. For instance. Eq. (12) can be optimized in arbitrary systems without reciprocal connectivity. This distinction makes Chomeo more general than alternative strategies acting on weight symmetry directly (Kolen & Pollack, 1994; Akrout et al., 2019).
4 EXPERIMENTS
In the following experiments, we used the setting of convergent recurrent neural networks (Ernoult et al., 2019), as well as the linear readout for optimizing the cross-entropy loss (Laborieux et al., 2021) (see appendix E.2). Simulations were implemented in JAX (Bradbury et al., 2018) and Flax (Heek et al., 2020) and datasets obtained through the Tensorflow Datasets API (Abadi et al., 2015). Our code is available on GitHub' and hyperparameters can be found Appendix E.3.
Holomorphic EP matches automatic differentiation computing dgu". We sought under- stand the amount of bias due to the finite nudge B when using different ways of estimating the neuronal error vector dou". To this end, we trained two-hidden layer network with independent forward and backward connections (Fig. 1b) on the Fashion MNIST dataset (Xiao et al., 2017) using stochastic gradient descent for 50 epochs. To investigate the evolution of weight symmetry during training, the forward and backward connections were initialized symmetrically. We compared different ways of estimating the neuronal error vector dguu. As the ideal "ceiling" case, we computed the ground truth using forward-mode automatic differentiation in JAX. Additionally, we computed the classic one- sides EP estimate (Scellier et al., 2018) denoted by "Classic" in Table 1, as well as the Cauchy integral estimate (Eq. 7) computed with various number N of points (see Appendix E. In all cases we ran simulations with two teaching amplitudes [@] = 0.05 and |3] = 0.5. We report the final validation errors in Table 1. We observed that the bias of the nudge is already noticeable low teaching amplitude when comparing to better estimates obtained using higher N or the ground truth. The bias became more noticeable when the teaching amplitude was increased. The lowest average validation error was obtained with N = 6 points and |3] 0.5, matching the ground truth within statistical uncertainty, and consistent with our theory. Finally, to test the potential of running hEP in continuous time, we implemented the continuous-time estimates of Eqs (8) and (9) where the average spanned five oscillation periods and N = 4 points were used. Crucially, there was no free phase in this setting. Despite longer numerical simulation times in this case (see Appendix E.4), we observed that the continuous-timg time estimate only suffered from moderate performance drop with respect the ground truth dyu" despite the complete absence of the free phase, thereby confirming our predictions.
'tttps:/ //gi thub. com httmm//aboroieuueeem /general zed-holos holo-ep

---

Published as a conference paper at ICLR 2024
Jacobian asymmetry leads to different learning dynamies. To see whether and how Jacobian asymmetry influences training, we studied the dynamics of the neuronal error vectors dgu", obtained through hEP, and ô from RBP respectively. We used the same experimental setting as in the previous paragraph and trained neural networks with varying degrees of initial weight asymmetry. The degree of initial weight asymmetry was controlled by setting the backward weights wy as w/ f- sin( (a)) w,, + cos c where Wy are the forward weights. Throughout training, we recorded the evolution of the angle a, and the cosine between both neuronal error vectors (Fig. 3).
We observed that networks trained with RBP all performed comparably in terms of loss (Fig, 3c) whereas their weight symmetry reduced quicker than networks trained with hEP (Fig. ba,b). Furthermore, RBP trained networks settled in a parameter regime in which the cosine between ó and dguu was negative for all except the output layer (Fig. 3e,f). contrast, the performance of networks trained with hEP strongly depended on the initial degree of symmetry (Fig. 3d), and although the cosine angle with RBP decreased during training, it remained positive throughout. These findings could suggest that learning with hEP alone reaches a parameter regime in which the cosine with ò is too low to further decrease the loss while maintaining the current performance level. In summary. networks that start with symmetric weights, lose this symmetry during training, which leads to increasing bias with respect to the ground -tuth neuronal error vector and impaired task performance. In the next paragraph, investigate whether and what extent the homeostatic loss introduced above (cf Eq. (12)) can mitigate this problem.
Table 1: Validation error in % + stddev on Fashion MNIST (n 5) for different values of [3].
Teach, amp.
Classic
N
N =
N 6
Continuous-
True dgu
[3| = 0.05
15.8 + 0.6
14.6 = 0.4
14.7 + 0.7
14.5 = 0.6
15.8 0.8
14.7 0.6
|6| 0.5
38.4 + 6.2
16.3 0.7
14.8 + 0.8
14.3 = 0.6
17.3 = 1.0
Training uses 6ниP
C)
e)
Initial d=0
Weight angle a (deg)
1.00
cast Bpu))
Output layer
0.75
Second layer
Pidden byyer
0.50
25
0.25
0.00
b)
Training uses dRu*
Initial g=0
Weight anglie (deg)
1.00
costomer Opti*)
75
0.75
50
Training
0.50
25
0.25
Epoch
Epoch
Epoch
Figure 3: Comparison of training dynamics of two-hidden-layer MLP Fashion MNIST using RBP (top row) and generalized hEP (bottom row). a,b) Evolution of the angle between forward and backward connections during training on for varying initial angle. Training tends to reduce weight symmetry. c,d) Learning curves. The training loss is dependent on initial angle only for generalized hEP. e,f) Evolution of the cosine similarity between neuronal error vectors of RBP and hEP over training. Curves are averaged over three seeds and shaded areas denote + 1 stddev.
Jacobian homeostasis improves functional symmetry and enables training on larger datasets. To study the effect of the homeostatic loss (Eq. (12)). we added it scaled by the hyper parameter Anomeo to the cross entropy loss (see Appendix E.3). For all experiments
estimated the average in Eq. (12) using 5 stochastic realizations of a Gaussian noise per sample in the mini-batch. With these settings, trained the same network in the previous paragraphs with the homeostatic loss
7

---

Published as a conference paper at ICLR 2024
a)
b)
0000
Validation arrer (8%) Valiation ertor (93
homeo
jacoblanssymiee
1.0
collivar Ago
1.00
Layer
0.8
0.75
Layer
Layer
e)
256
256
0.6
0.25
1.0
1.00
coolirg Aet""
Loya #2
0.75
Lono
jacobian
i) 256
256
j)
k)
Epoch
Valiation ertor (%9
1.00
Weights anple
0.75
44
Layer
40
0.50
Laver 92
#1
(16x16)
(8x8)
(4x4)
(1x11
20-
0.25
25
50
75
25
50
75
25
75
Epoch
Epoch
Figure 4: The homeostatic loss improves hEP training of arbitrary dynamical systems by acting directly on the Jacobian. Each row corresponds to a specific architecture-datsse pair. a) Multi-layer architecture where layers are reciprocally connected. Evolution of the validation error b), Jacobian symmetry measure c), and layer-wise cosine between neuronal error vectors of hEP and RBP d) during training of Fashion MNIST. e-h) Same plots as a-d for an architecture where the output layer directly feeds back to the first layer. i) Recurrent convolutional architecture used CIFAR-10. Evolution of validation error j). angle between weights k) and layer-wiss wise cosine between the neuronal error vectors of hEP and RBP I). Curves are averaged over five random seeds for Fashion MNIST and three for CIFAR-10 10 and shaded areas denote standard error.
on Fashion MNIST (Fig. 4a) and found a small reduction in validation error. We further observed that the added homeostatic loss increased the symmetry of the Jacobian, as defined by the symmetry measure STI (IIIIII + ||[A|]]) (Section 3), over the course of training. In line with this observation, the alignment between 5 and d (Fig. 4c-d) also increased for each layer.
Because the homeostatic loss is defined on the Jacobian instead of the weights, we hypothesized that its effect may more general than merely increasing weight symmetry. To test this idea, we introduced a network architecture without feedback connections between adjacent layers, thereby precluding the possibility for symmetric feedback. Specifically, we trained a multi-layer architecture in which the output units fed back to the first hidden layer directly (Fig. 4e) (Kohan et al., 2018; 2023; Dellaferrera & Kreiman, 2022). After training this antisymmetric architecture, we observed the same key effects on the validation error, Jacobian symmetry, and error vector alignment (Fig. 4g.h) in the reciprocally connected network, thereby confirming our hypothesis that the homeostatic loss is more general than weight alignment (Kolen & Pollack, 1994). In addition, we show in Appendix D that the homeostatic loss generalize to predictive coding networks (Whittington & Bogacz, 2019).
Since, the improvement in validation error was measurable, but small in the network architectures we studied (Fig. 4b,f), we wondered whether the positive effect of the homeostatic loss would be more pronounced on more challenging tasks. To this end, we extended a recurrent convolutional architecture (Fig. 4i; Laborieux & Zenke, 2022) to asymmetric feedback weights (see Appendix E.2 for further details). We trained this architecture using generalized hEP on CIFAR- 10, CIFAR-100 (Krizhevsky, 2009) vision benchmarks both with and without homeostatic loss.
Without the homeostatic loss, the network trained on CIFAR- reached 60.4% validation accuracy (Table 2). With the homeostatic loss, the performance increased to 84.3 (Table 2), with only approximate weight symmetry (Fig. 4k). Strikingly, this is only a reduction by 4.3% accuracy in comparison to the symmetric architecture (Laborieux & Zenke, 2022). When using N = 2 instead of the ground truth for estimating dyuu, we observed an additional drop by 2.9% points due to the finite nudge bias. We also noticed that the homeostatic loss has measurable effect on performance when training the asymmetric network with RBP (Table 2), suggesting that the homeostatic loss is only beneficial while not restricting model performance. Finally, reproduced similar findings on
8

---

Published as a conference paper at ICLR 2024
Table 2: Validation accuracy of asymmetric networks trained with generalized hEP (ours) and implementation of RBP. All values are averages (n 3) - stddev.
CIFAR-10
CIFAR-100
ImageNet 32 32
Top-1 (%)
Top-1 (%)
Top-5 (%)
Top-1 (%)
Top-5 (%)
hEP w/o hhomeo
60.4 + 0.4
35.2 0.3
64.4 + 0.5
hEP, N [8
81.4 0.1
51.1 = 0.8
79.2 + 0,4
hEP (True dgu")
84.3 0.1
53.8 + 0.8
81.0 + 0.3
31.4 + 0.1
55.2 + 0.1
RBP
87.8 + 0.3
60.8 + 0.2
84.4 0.1
RBP w/o Lhomeo
87.7 + 0.2
61.0 0.2
85.2 + 0.3
Sym. hEP N 211811
88.6 0.2
61.6 = 0.1
86.0 + 0.1
36.5 = 0.3 60.8 0.4
CIFAR-100 100 and ImageNet 32 x 32, with a remaining 5% gap on Top-5 validation accuracy from the perfectly symmetric architecture. Together these findings suggest that the addition of homeostatic objectives is increasingly important for training on larger datasets and further highlights the need additional homeostatic processes in the brain (Zenke et al., 2017; Stock al., 2022).
DISCUSSION
We have studied a generalized form of EP in the absence of perfect weight symmetry for finite nudge amplitudes. We found that relaxing the strong algorithmic assumptions underlying EP rapidly degraded its ability to estimate gradients in larger networks and on real-world tasks. We identified two main sources of bias in generalized EP (Scellier al., 2018) with adverse effect on performance: The finite nudge amplitude and the Jacobian asymmetry. Further, we illustrate how both issues can be overcome through oscillations by combining holomorphic EP (Laborieux & Zenke, 2022) with new form of Jacobian homeostasis that encourages functional symmetry, but without imposing strict weight symmetry. Finally, we show that our strategy allows training deep dynamical networks without perfect weight symmetry ImageNet 32 x 32 with only small gap in performance to ideal symmetric networks.
The role of weight asymmetry for computing gradients has received lot of attention (Kolen & Pollack, 1994; Lillicrap et al., 2016; Launay et al., 2020; Nøkland, 2016; Payeur et al., 2021; Greedy et al., 2022). Our work corroborates previous findings showing that learning with fully asymmetric weights results in poor alignment with back propagation which limits it to small datasets. Reminiscent of work Feedback Alignment (Lillicrap et al., 2016), training becomes networks and on larger datasets (Xiao et al., 2018; Liao et al.. 2016; Bartunov et al., 2018). Although our findings do allow completely dispensing with the weight symmetry requirement, they are more general since Jacobian symmetry and weight symmetry are not the same, a realization that may prove useful for future algorithmic developments.
challenging in deeper
Nevertheless, several questions pertaining to plausibility remain open. For instance, removing the bias with hEP requires the neurons to oscillate in the complex plane, which restricts its use oscillation- based learning and makes its biological interpretation challenging. It could be implemented through phase coding (Frady & Sommer, 2019; Bybee al., 2022), which would require an additional fast carrier frequency, suggesting potential links to the beta and gamma rhythm in neurobiology (Engel et al., 2001). Another limitation of EP is the convergence to an equilibrium. This requirement makes EP costly digital computers, which simulate the substrate physics, whereas analog substrates could achieve this relaxation following the laws of physics (Yi et al., 2023; Kendall et al., 2020).
In summary, our work further bridges the gap between EP's assumptions and constraints found in physical neural networks, and opens new avenues for understanding and designing oscillation- learning algorithms for power efficient learning systems.

---

Published as a conference paper at ICLR 2024
ACKNOWLEDGMENTS
We thank all members of the Zenke Lab for comments and discussions. We thank Maxence Ernoult, Nicolas Zucchet, Jack Kendall and Benjamin Scellier for helpful feedback. This project was supported by the Swiss National Science Foundation Igrant numbers PCEFP3 202981 and TMPFP3 TMPFP3_2I0022 EU's Horizon Europe Research and Innovation Programme (CONVOLVE, grant agreement number 101070374) funded through SERI (ref 1131-52302) and the Novartis Research Foundation. The authors declare no competing interests.
SSFFP32210222 210282],
REFERENCES
Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, llya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaogiang Zheng. TensorFlow: Large-scale machine learning heterogeneous systems, 2015. URL http: http://ennnno....eeee//coo org/ Software available from tensorflow.org.
Mohamed Akrout, Collin Wilson, Peter Humphreys, Timothy Lillicrap, and Douglas Tweed. Deep Learning without Weight Transport. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
Luis B Almeida. A learning rule for asynchronous perceptrons with feedback in combinatorial environment. Artificial neural networks: concept learning, pp. 102-111. IEEE Computer Society Press, 1990.
Vidyesh Rao Anisetti, Ananth Kandala, Benjamin Scellier, and JM Schwarz. Frequency propagation: Multi-mechanism learning in nonlinear physical networks. arXiv preprint TrXiv:228800886 2022.
Walter Appel. Mathematics for physics and physicists. Princeton University Press, 2007.
Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Deep equilibrium models. Advances in Neural
Information Processing Systems, 32, 2019.
Shaojie Bai, Vladlen Koltun, and Zico Kolter. Stabilizing equilibrium models by jacobian regular- ization. arXiv preprint arXiv:2106. 14342, 2021.
Pierre Baldi. Gradient descent learning algorithm overview: A general dynamical systems perspective. IEEE Transactions on neural networks, 6(1):182-195, 1995.
Pierre Baldi and Fernando Pineda. Contrastive learning and neural oscillations. Neural computation, 3(4):526-545, 1991.
Sergey Bartunov, Adam Santoro, Blake Richards, Luke Marris, Geoffrey E Hinton, and Timothy Lilli- crap. Assessing the scalability of biologically-motivaeed deep learning algorithms and architectures. Advances in neural information processing systems, 31, 2018.
Irem Boybat, Manuel Le Gallo, SR Nandakumar, Timoleon Moraitis, Thomas Parnell, Tomas Tuma, Bipin Rajendran, Yusuf Leblebici, Abu Sebastian, and Evangelos Eleftheriou. Neuromorphic computing with multi-memristive synapses. Nature communications, 9(1):2514, 2018.
James Bradbury, Roy Frostig. Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman- and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. URL http:// thub. com/goog! leooolll//e jax.
Connor Bybee, Paxon Frady, and Friedrich T Sommer. Deep learning in spiking phasor neural networks. arXiv preprint 204002220 2022.
10

---

Published as a conference paper at ICLR 2024
David G. Clark, L. F. Abbott, and Sue Yeon Chung. Credit Assignment Through Broadcasting Global Error Vector, October 2021.
Claudia Clopath and Wulfram Gerstner. Voltage and spike timing interact in stdp-a a unified model. Frontiers in synaplic neuroscience, 2:25, 2010.
Yang Dan and Mu-ming Poo. Spike timing-dependent plasticity of neural circuits. Neuron, 44(1): 23-30, 2004.
Corentin Delacour, Stefania Carapezzi, Madeleine Abernot, Gabriele Boschetto, Nadine Azemard, Jeremie Salles, Thierry Gil, and Aida Todri-Sanial. Oscillatory neural networks for edge ai computing. In 2021 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), pp. 326-331. IEEE, 2021.
Giorgia Dellaferrera and Gabriel Kreiman. Error-driven Input Modulation: Solving the Credit Assignment Problem without Backward Pass. In Proceedings of the 39th International Conference on Machine Learning, pp.
PMLR, June 2022.
Stefan Elfwing, Eiji Uchibe, and Kenji Doya. function approximation in reinforcement learning. Neural Networks, 107:3-11, 2018.
linear units for neural network
Andreas K Engel, Pascal Fries, and Wolf Singer. Dynamic predictions: oscillations and synchrony in top-down processing. Nature Reviews Neuroscience, 2(10):704-716, 2001.
Maxence Ernoult, Julie Grollier, Damien Querlioz, Yoshua Bengio, and Benjamin Scellier. Updates of equilibrium prop match gradients of backprop through time in an mnn with static input. Advances in neural information processing systems, 32, 2019.
Maxence Ernoult, Julie Grollier, Damien Querlioz, Yoshua Bengio, and Benjamin Scellier. Equilib- rium propagation with continual weight updates. arXiv preprint wwXiv:2005.04163 2020.
Maxence Ernoult, Fabrice Normandin, Abhinav Moudgil, Sean Spinney, Fugene Belilovsky, Irina Rish, Blake Richards, and Yoshua Bengio. Towards scaling difference target propagation by learning backprop targets. arXiv preprint arXiv:2201.134155 2022.
E Paxon Frady and Friedrich T Sommer. Robust computation with rhythmic spike patterns. Proceed- ings of the National Academy of Sciences, 116(36): (06(66))18050-110 8050-18059, 2019.
Charlotte Frenkel and Giacomo Indiveri. Reckon: A 28nm sub-mm2 task- spiking recurrent neural network processor enabling on-chip learning over second-long timescales. In 2022 IEEE International Solid-State Circuits Conference (ISSCC), volume 65, pp. 1-3. IEEE, 2022.
Will Greedy, Heng Wei Zhu, Joseph Pemberton, Jack Mellor, and Rui Ponte Costa. Single-phase deep learning in cortico-cortical networks. Advances in Neural Information Processing Systems, 35:24213 242255, 2022.
Stephen Grossherg. Competitive learning: From interactive activation to adaptive resonance. Cogni- live science, 11(1):23-63, 1987.
Manu Srinath Halvagal and Friedemann Zenke. The combination hebbian and predictive plasticity learns invariant object representations in deep sensory networks, bioRxiv, 2022.
Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Mare van Zee. Flax: A neural network library and ecosystem for JAX, 2020. URL http:/ / thub. cww/ e/ flax.
Geoffrey Hinton. The forward-forward algorithm: Some preliminary investigations. arXiv preprint arXiv:2212.13345, 2022.
Rasmus Høier, D. Staudt, and Christopher Zach. Dual propagation: Accelerating contrastive hebbian learning with dyadic neurons. In Proceedings of the 40th International Conference Machine Learning. volume 202 Proceedings of Machine Learning Research, pp. 13141-13156. PMLR, 23-29 Jul 2023. URL ht https:ttt/t. hetp::/ppoocceeee ngs ml www.ml.....///eiemm wttpprss/////2/hiieieeeee htwp:////h//ee22/ html.
/

---

Published as a conference paper at ICLR 2024
John J Hopfield. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the national academy of sciences, 81(10):3088- 3092, 1984.
Michael F Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian and Computation, 18(3):1059-1076
smoothing splines. Communications 1989.
Giacomo Indiveri, Bernabé Barares-Barr,nco, Tara Julia Hamilton, André van Schaik, Ralph Etienne- Cummings, Tobi Delbruck. Shih-Chii Liu, Piotr Dudek, Philipp Häfliger, Sylvie Renaud, et al. Neuromorphic silicon neuron circuits. Frontiers in neuroscience, 5:73, 2011.
Adrien Journ,, Hector Garcia Rodriguez, Qinghai Guo, and Timoleon Moraitis. Hebbian deep learning without feedback. In The Eleventh International Conference on Learning Representations, 2023. URL https: wwwpreview.ee....../.. foott/oorrmmmmdd_d__tt
forum?
Jack Kendall, Ross Pantone, Kalpana Manickavasagam, Yoshua Bengio, and Benjamin Scellier. Training end-to-end analog neural networks with equilibrium propagation. arXiv preprint Sttv:2006.0098//1 2020.
Adam Kohan, Edward A. Rietman, and Hava T. Siegelmann. Signal Propagation: The Framework for Learning and Inference Forward Pass. IEEE Transactions on Neural Networks and Learning Systems, pp. 1 1-12, 2023. ISSN 2162-237X, 2162-2388.
doi: 10.1 2.1109/TNNISS
Adam A. Kohan, Edward A. Rietman, and Hava T. Siegelmann. Error Forward- wwward-Propagation Reusing Feedforward Connections to Propagate Errors in Deep Learning, August 2018.
John Kolen and Jordan B Pollack. Backpropagation without weight transport. In Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94), volume 3. pp. 1375-1380. IEEE, 1994.
Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.
Axel Laborieux and Friedemann Zenke. Holomorphic equilibrium propagation com- putes exact gradients through finite size oscillations. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural Infor- mation Processing Systems, volume 35, pp. 12950-12963. Curran Associates, Inc., saaps_fleesspppp//22a/1 ptppp/ppppp/////////
2022. URL https: httt:::s/prrceedinngg neurips cc/paper files/lll//lell 4tlee/555a114e655ffee ddeee9925bbbaaa wew.aaaa000eeeee00eeee0eee Seeeeefceee-eeeeeeee Paper- Conference pdf..
Axel Laborieux, Maxence Erouut,, Benjamin Scellier, Yoshua Bengio, Julie Grollier, and Damien Querlioz. Scaling equilibrium propagation to deep convnets by drastically reducing its gradient estimator bias. Frontiers neuroscience, 15:129, 2021.
Julien Launay, lacopo Poli, François Boniface, and Florent Krzakala. Direct feedback alignment scales to modern deep learning tasks and architectures. Advances in neural information processing systems, 33:9346-9360, 2020.
Lee, Saizheng Zhang, Asja Fischer, and Yoshua Bengio. Difference target propagation. In Joint european conference on machine learning and knowledge discovery in databases, pp. 498-515, Springer, 2015.
Qianli Liao, Joel Z. Leibo, and Tomaso Poggio. How Important Weight Symmetry in Backpropa- gation?, February 2016.
Renjie Liao, Yuwen Xiong, Ethan Fetaya, Lisa Zhang, KiJung Yoon, Xaq Pitkow, Raquel Urtasun, and Richard Zemel. Reviving and improving recurrent back prop-propggttion. In International Conference on Machine Learning, pp. 3082-3091. PMLR, 2018.
Timothy P Lillicrap, Daniel Cownden, Douglas B Tweed, and Colin J Akerman. Random synaptic feedback weights support error backpropagation for deep learning. Nature communications, 7(1): 1-10, 2016.
12

---

Published as a conference paper at ICLR 2024
Timothy P Lillicrap. Adam Santoro, Luke Marris, Colin J Akerman, and Geoffrey Hinton. Backprop- agation and the brain. Nature Reviews Neuroscience, 21(6):33--34, 2020.
David Lipshutz, Charles Windolf, Siavash Golkar, and Dmitri Chklovskii. A Bi- ologically Plausible Neural Network for Slow Feature Analysis. In Advances in Neurul Information Processing Systems, volume 33, pp. 14986-14996. Curran Asso-
ciates, Inc., 2020. URL https: http:ttttttttsseeeeeeeee http:///ooccedeeeeeeee ier.innni.nnnneeneeeeeeee neurips cctpaper//020/ a07315 abcCaac Abstract.htm
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint
arXiv:1711.05101, 2017.
Sindy Löwe, Peter O'Connor, and Bastiaan Veeling. Putting end end-to-end: Gradient-isolated learning of representations. Advances in neural information processing systems, 32, 2019.
Alexander Meulemans, Francesco Carzaniga, Johan Suykens, João Sacramento, and Benjamin F Grewe. A theoretical framework for target propagation. Advances in Neural Information Processing Systems, 33:20024 20036, 2020.
Alexander Meulemans, Matilde Tristany Farinha, Javier Gareía Ordóñez, Pau Vilimelis Aceituno, João Sacramento, and Benjamin F Grewe. Credit assignment in neural networks through deep feedback control. Advances Neural Information Processing Systems, 34:4674-4687 2021.
Arild Nøkland. Direct feedback alignment provides learning in deep neural networks. Advances in neural information processing systems, 29, 2016.
Alexandre Payeur, Jordan Guerguiev, Friedemann Zenke, Blake A Richards, and Richard Naud. Burst- dependent synaptic plasticity coordinate learning in hierarchical circuits. Nature neuroscience, 24(7):1010-109, 2021.
Fernando Pineda. Generalization of back propagation to recurrent and higher order neural networks. In Neural information processing systems, 1987.
Maria Refinetti, Stéphane d' Ascoli, Ruben Ohana, and Sebastian Goldt, Align, then memorise: The dynamics of learning with feedback alignment, June 2021.
Mengye Ren, Simon Korbllith, Renjie Liao, and Geoffre Hinton. Scaling forward gradient with local losses. In The Eleventh International Conference Learning Representations, 2023. URL https: http://///peeeeeeeeeeeeee net/ forum? ? i d= JxpBP1JM15
David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. Nature, 323(6088):533-536, 1986.
Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. JEEE transactions on neural networks, 20(1):61-8)) 2008
Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap between energy- based models and backpropagation. Frontiers in computational neuroscience, 11:24, 2017.
Benjamin Scellier, Anirudh Goyal, Jonathan Binas, Thomas Mesnard, and Yoshua Bengio. Gener- alization of equilibrium propagation to vector field dynamics. arXiv preprint srXiv:1808.04873 2018.
Catherine D Schuman, Thomas E Potok, Robert M Patton, J Douglas Birdwell, Mark E Dean, Garrett Rose, and James S Plank. A survey of neuromorphic computing and neural networks in hardware. arXiv preprint arXiv: 1705506963 2017.
A Stergiou, R Poppe, and G Kalliatakis. Refining activation downsampling with softpool. arxiv 2021. arXiv preprint arXiv:2101.0000 004400, 2021.
Christopher H Stock. Sarah E Harvey, Samuel A Ocko, and Surya Ganguli. Synaptic balancing: A biologically plausible local learning rule that provably increases neural network noise robustness without sacrificing task performance. PLOS Computational Biology, 1809)e1010418, 2022.
13

---

Published as a conference paper at ICLR 2024
Chetan Singh Thakur, Jamal Lottier Molin, Gert Cauwenberghs, Giacomo Indiveri, Kundan Kumar, Ning Qiao, Johannes Schemmel, Runchun Wang. Elisabetta Chicca, Jennifer Olson Hasler, et al. Large-scale neuromorphic spiking array processors: A quest mimic the brain. Frontiers in neuroscience, 12:891, 2018. 2.
James CR Whittington and Rafal Bogacz. Theories of error back-propagation in the brain. Trends in cognitive sciences, 23(3):235-255 2019.
Ezekiel Williams, Colin Bredenberg, and Guillaume Lajoie. Flexible phase dynamics for bio- dio-plausible contrastive learning. arXiv preprint 2.XKv:2300.1224311 2023.
Laurenz Wiskott and Terrence Sejnowski. Slow feature analysis: Unsupervised learning of invariances. Neural computation, 14(4):715-770,22) 2002.
Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747 2017.
Will Xiao, Honglin Chen, Qianli Liao, and Tomaso Poggio. ioologically-plausibe learning algorithms can scale to large datasets, December 2018.
Su-in Yi, Jack D Kendall, R Stanley Williams, and Suhas Kumar. Activity- training of deep neural networks using memristor crossbars. Nature Electronics, 6(1):45-51, 2023.
Friedemann Zenke, Wulfram Gerstner, and Surya Ganguli. The temporal paradox of Hebbian learning and homeostatic plasticity. Curr Opin Neurobiol, 43:166-176, April 2017. ISSN 0959-4388. doi: 10. 10.1016ff.comb222110...0011 URL http: /twp//wwww.//./////////////.// wwwp:/sccienedireeee.... rectt. com/ wtwp/cccc//aa.eee.eeeo. sscsnccc/c//////////// article/ps1//99544881300099
14

---

Published as a conference paper at ICLR 2024
A MATHEMATICAL BACKGROUND
A.1 COMPLEX ANALYSIS
Here we provide intuitions and the minimal theoretical background on complex analysis to allow appreciating the results presented in Sections 3.1 and 3.2. We refer the interested reader to (Appel, 2007) for a more complete introduction.
The notion of complex differentiability in point 2g E C defined for a function f f:2 C f()) C by the existence of the limit:
f(z) f(zo)
lim 22
(13)
2 20
The limit is written f'(z)), and the function / is said be "holomorphic" in 20- Although the definition is similar to R- 2--difereetiiabliit, the fact that the limit is taken with complex number that can go to 2g from all directions in the complex plane makes the variation of more "rigid". However, most usual functions on real numbers can be extended to complex inputs and outputs and are holomorphie, such as the exponential, the sigmoid, all polynomials, trigonometric functions, softmax, and logarithm. Non-holomorphic functions include functions that, for instance, use the absolute value or min/max operators.
In exchange for the rigidness of holomorphic functions, one obtains the possibility to express their derivatives at any order through integrals as described by the Cauchy formulas.
f(2) 2im Sr (z (2-20)nt 20) dz,
n!
(z))
(14)
where nf is the factorial product of integers up n. is the imaginary unit, and the integral is taken over path C C going around zg once and counterclockwise, and provided that f is holomorphic on set that includes the path. For instance, for n and the path y 0 E [- т] + 2g + re doing the change of variable de = rie d6 yields the mean value property linking the value of f in 2g its variation on circle of radius going around zn:
1 ["
f'z0 + re)
f(=0) =
ric
2iv 20 rei0 20
1
f(z0)
f(0 + re )d0
(15)
2m
And f'(zo) can be computed by plugging n = in Eq. (14), and the same change of variable:
f(z)
f'(zo)
2im Jn (a 20) ) dz,
f(20 re(0)
f'(zo)
(2o + rei0 20) ric
de,
2im
f'(z))
f(20 + re, d0.
(16)
2
Importantly, these formulas are integrals over a non-vanishing radius r > 0 where the integrands only involve the function S. which is promising conceptual step toward biological plausiblity and hardware design since integrals are easier to implement in noisy substrates than finite differences.
A.2 HUTCHINSON TRACE ESTIMATOR
The Hutchinson trace estimator (Hutchinson, 1989) refers to the following "trick" to estimate the trace of square matrix M [m:y]: suppose have random vector IR. such that Elee T,
15

---

Published as a conference paper at ICLR 2024
where I is the identity matrix. Then we have:
El
Me]
E
Σ mijCrey
Σ if
may [EYEy]
Σ
mislis
ij
Σ m4 = (r(M).
(17)
Since the Frobenius norm can be expressed with the trace operator, can be estimated in the same way. For instance, the Frobenius norm of the asymmetric part A of the Jacobian Jr, A {)rr - J
given by: || Tr(ATA) A)
Tr ((J) - Jr) JF))
4
1
Tr (JPJs JJJJ JEJF JFJF)
4
1
TT(FFJE))
1
Tr((z)
2
2
1
Ele JJ
Elee J
using Eq. (17)
2
2
[[] - JT.s].
(18)
2
To obtain the fourth line, we used the linear property of the trace operator, the fact that the trace is unchanged by the matrix transposition, and fact that (r(M, Tr(M2M1 for any two matrices M1 and My. which is a consequence of the symmetry of the Frobenius inner product. The final quantity in Eq. (18) can estimated efficiently with vector-Jacobian product routines available in automatic differentiation frameworks such as JAX or PyTorch, without instantiating the full Jacobian matrix.
B RELATION BETWEEN THE ENERGY AND VECTOR FIELD GRADIENT FORMULAS
Here we briefly provide an intuition for how the EP gradient formula obtained with the energy function formalism (Scellier & Bengio, 2017) relates to the EP gradient formula obtained directly with the vector field (Eq. (6), Scellier et al. (2018)). Although the formulas seem different, they are in fact equivalent. The main takeaway is that the connections Wi and wy are fused into a single parameter in the energy function formalism, whereas the vector field formalism distinguishes the two. For continuous Hopfield model with two-body interaction terms of the form (w,, (u:))wi o (u;), the energy-based (EB) gradient of the loss with respect Wiy is (Scellier & Bengio, 2017):
dc
do ( )o(()
dwij
d3
EB
8=0
If use the product rule of derivatives, we obtain:
dL
duf
duf d3
= of
+ (uggoo'()
dw
d3
EB
1==0
8=0
Both terms in the sum corresponds respectively to the vector field (VF) gradient formula for way and (u;)) Therefore, we have:
Wjt. Here OF
Jwiy
dc
dc dl
dwns
dwis
duyz Iv
lBB
IvF
16

---

Published as a conference paper at ICLR 2024
C
THEORETICAL PROOFS
C.I CONVERGENCE OF THE CONTINUOUS-TIME ESTIMATE
We recall the limit:
1 t|
[ u 2 dr
du"
8-+00 d3 o==
We split the running time / into an integer & amount of periods = kT + Then by splitting the integral we have:
u
2i/T/T dr
1
J
KT+
(g(r)e
-irt/T dr
[33
1/3
2i/T/T + So" uBcc) -ige--ttrrrr dr
k
ug(r)e
t|
1
du"
+ L e 2ier//// dr
|T [8]
kT
t|
d3
8=0
1 l'
du
"z(r)
TT dr.
(19)
t
dB
8=0
t|
Since 0 < 1' < T, the first term converges toward deuu, and the second term is a bounded integral divided by t, and therefore goes to as t + oc. For the pre-synaptic term recall the statement:
/
OF 00
OF
( dr
00 (u))-
The proof is the same above, once we have shown that:
1
OF d0 (dT
OF
00 (u))-
(20)
By applying the Mean Value Theorem (Eq. (15)) (Appel, 2007) to the function yp B E D.,
3e (ug) in 3 = O, we have that:
OF 00
OF
(u ) da,
2m
00
where is radius such that the circle is contained in the domain of definition Du- of yp. Then apply the equation in r = | and do the change of variable a = 2 which gives Eq. (20).
C.2 BIAS TERM FROM JACOBIAN ASYMMETRY
We recall that dou" and are linked by:
dsu" (r 8.
(21)
Let S and A be the symmetric and kymmetriet parts of Jr. such that = (Jr(un)
Jr (T(u))T T)2 and A (Jr(u)) Jr(uj) 1/22. Since Jr Ju(u)) is assumed invertible and ||ll
17

---

Published as a conference paper at ICLR 2024
d)
Volidation (%)
25
lethout nomeo 6s5
Jacobian opmmetry
Homeostatic loss
with homes Ioss
0.190
20
0.185
15
0.180
10
256
256
10
10 Epach
10
10
Epoch
Epoch
Figure 5: The same experiment Fig. 4, but run on a Predictive Coding Network (Whittington & Bogacz, 2019). a) Architecture where each layer consists of value neurons (circles) and error neurons (squares). Evolution during training the validation error b), Jacobian symmetry measure c), and Homeostatic loss d).
assumed small, we have that S is invertible and:
(J (u T = (S + ()
(T ( --+S-1A) -- A)
(I (--S-AA) (1- -S-1A
- () A)} (1-5-1A)
(-1) AA)
(-))((5-aa))t) 1/ht1
k=
k=0
Σ
(-1)*(S-1 1A)* +
(-1)*(S-2A)
k=
k=1
I 2S A+2 Σ (-1) A)*
We used the Neumann Series compute the inverse (I + 1-1 A) which converges given the
assumption |||s~|A|| AAll + O. From Eq (21) we can write:
douu = ò A8 + o(S
D APPLICATION TO PREDICTIVE CODING NETWORKS
To demonstrate that our approach of regularizing the Jacobian to improve its symmetry is not restricted to the networks trained in the main manuscript, and generalize to other dynamical systems, we run an additional training experiment on a different model architecture, namely predictive coding networks (PCNs) (Whittington & Bogacz, 2019).
The key feature of the predictive coding network architecture is the presence in each layer of explicit error neurons e in addition to value neurons u, (Fig. 5). The evolution of the error and value neurons are given by:
El u; w/ (((-))))
du ei + ('u) O wf411
dt
where 3 denotes the element-wise product, and the superscripts f and denote forward and backward weights respectively. We train such a network with generalized hEP Fashion MNIST with and without the homeostatic loss, and observe the same behavior as in Fig. that optimizing the homeostatic loss for the PCN improves the symmetry of the Jacobian of the network, which translates into better validation error (Fig. 5). The hyperparameters used this experiment are reported in Table 3.
18

---

Published as a conference paper at ICLR 2024
E SIMULATION DETAILS
E.1 NUMERICAL ESTIMATE OF THE CAUCHY INTEGRAL
For evaluating the integral in Eq. (7), we an integer number of N values B
Bk | 2in
k [0,1, N N- 1].
The integral is then approximated by:
N N
1 T
Birr / dT ~d
u
kinkNN
(22)
k=0
The leading term of the bias when using N points is |/| /(N 1)!, where is the factorial product. However, unless specified we used forward mode automatic differentiation to compute the ground truth dgu" without bias.
E.2 ARCHITECTURE AND DYNAMICS
Multi-layer networks. The disrete dynamics of the multi-layer networks of experiments in Figs. 2,
3, on Fashion MNIST read:
U1 Win' Er in (i.ro(ur) b
u Er (u bi,
((LL-1((UL-1) by Sw (y softmax (Wroo bro)),
uL
where Wi,1' is the weight matrix going from layer l' to I, and by the bias of layer l. (Wros bro) is 10 x 10 readout layer of the parameterized cross-entropy loss of Laborieux et al. (2021). In partieular, does not belong to the network connections since does not influence the free dynamics (when 0). The activation is the shifted sigmoid + 1/(1 e e44m22).
Convolutional networks. For the convolutional neural network experiments, the discrete dynamics read:
P(w( x) + P(u1, w². u2) + b1,
+ P(w- o(u1)) + p(uz, w&. uz) + b - P(wf * (u2)) ) + p(u3, w us) + b3,
u4 f P(w/ * (u3)) + b4 + Bw (y softmax (wroo(ur.) + bya)).
Where the superscripts f. respectively mean forward and backward weights. denotes Softmax pooling (Stergiou et al., 2021). The backward convolutional module going from 2441 to un is defined as:
P(u, w², 11+1
[w-1
P(w" ouu)))
(w))
The activation is a weigoid-weighted linear unit (Elfwing et al., 2018) defined by:
Ite- (1
2 2 1+2-
(x)
The shapes of u U22 Uk, u are respectively (16, 16, 128). (8,8, 256), (4,4,512), (1,1,11)), The weights are all 3 x 3 kernel with no strides. The paddings are all 'same" except for the last layer. The Softmax pooling has stride and window-size
It is defined by (Stergiou et al., 2021):
Σ
V/
Σ
Ejen
e
Eis
where R. is the 2 window.
19

---

Published as a conference paper at ICLR 2024
E.3 HYPERPARAMETERS
Choice of hyperparameters. The learning rate was the main hyperparameter searched for by coarse grid search. The number of time steps tuned such that the networks have enough time equilibrate by measuring the norm of the vector field. The coefficient for the homeostatic loss was searched by grid search over the range 0.1, 1.0, and 10.0, Hyperparameters for the CIFAR-I0// and ImageNet 32 x 32 were chosen based on those reported in the literature for similar experiments (Laborieux al., 2021; Laborieux Zenke, 2022).
Table 3: Hyperparameters used for the Fashion MNIST training experiments.
Hyperparameter Fig. 3, Table 1, Fig. 4e-h)
Fig. 4a-d)
Fig. 5
Batch size Optimizer
50
50
50
SGD
Adam
SGD
Loss
X-ent.
X-ent.
Squared Error
Learning rate Momentum
le-2
le-4
le-2
0.9
0.9
Epochs
50
50
20
Ahomeo
0.1 (Fig. 4e-h)
1.0
1.0
T free
150
150
50
T nudge
20
In the case of continuous- time estimates (Table 1, Eqs. (8), (9)), used 60 time steps for each of the N = 3 values, and five periods, for a total to 1200 time steps per batch.
Table 4: Hyperparameters used for the CIFAR-10, CIFAR-100 and ImageNet 32 x 32 training
experiments.
Hyperparameter
Fig. 4i-1)
Table 2 CIFAR-(10/100)
Table 2 ImageNet 32 x 32
Batch size
256
256
256
Optimizer
SGD
SGD
SGD
Learning rate
5e-3
7e-3
5e-3
Weight decay
le-4
le-4
le-4
Momentum
0.9
0.9
0.9
Epochs
90
150
90
Ahomeo
10 or
Tire
250
250
250
T Tuudge
40
40
40
Learning rates were scaled layer-wise by [25, 15, 10, 8. 5] and decayed with cosine schedule (Loshchilov & Hutter, 2017).
E.4 HARDWARE AND SIMULATION TIMES
single RTX 5000 GPU, for 10mins per and 1 hour for the continuous-timg estimate. The convolutional network simulations were run on an in-house cluster consisting of 5 nodes with 4 v100 NVIDIA GPUs each, node with 4 A100 NVIDIA GPUs, and one node with 8 A40 NVIDIA GPUs. Runs on 1IFAR-10/100 took about hours each, and the Imagenet 32 32 splitting the batch. The free phase with the convolutional architecture was done in half precision, which provided a 2x speed up compared using full precision.
Simulations Fashion MNIST
took 72 hours each. Each run was parallelized over 4 or 8 GPUs by
20


Reviews:
[{"review_id": "mbGGskEW3A", "reviewer": "Reviewer_EQJi", "summary": "Equilibrium propagation, an alternative to backprop requires weight symmetry and nudges to yield unbiased gradient estimates. Generalization of Equilibrium proportion to non-symmetric dynamical system exists but shown to work only for simple problems.  \nThe paper proposes an extension of holomorphic EP to non-symmetrical dynamical systems and shows good results across different vision benchmarks.", "questions": "No further questions as such. The paper has good results across different vision benchmarks.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Analysis of bias in gradient estimation is helpful to the reader. \n- Incorporating functional symmetry through the use of matching jacobians is an interesting idea. It’s similar to reconstruction error term used in methods like Target Propagation. Here, authors optimize the homeostatic loss with respect to all the weights as compared to using only feedback weights.", "weaknesses": "- The paper is generally well written, though introduction is a bit complex if the reader is not aware of previous work (holomorphic EP).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "HpCiizyzcQ", "reviewer": "Reviewer_8uWs", "summary": "_Disclaimer: I have reviewed this paper recently again at a different venue. It has since been revised slightly. Some of my comments still apply to this version and are copied here verbatim from my previous review._\n\nThe paper focuses on Equilbrium Propagation (EP), i.e. a learning algorithm for neural networks that was introduced relatively recently and aspires to be more biologically plausible and more suitable for analog hardware than back-propagation (BP). The theoretical version of EP relies on an infinitesimal perturbation (which is not feasible to achieve in physical settings) and on symmetric weights (which are also unrealistic in biology and constrain hardware design). The paper aims to understand the impact of each of these two issues separately. It achieves to do so in the \"holomorphic EP\" (hEP) setting, i.e. under the assumption of complex-valued neurons. The authors achieve this by extending the theory of hEP to asymmetric connectivity, where they show that it is possible to obtain exact estimates of the gradient despite finite perturbations, therefore suggesting that the bigger issue is weight asymmetry. They then proceed to tackle the problem of asymmetry by introducing a loss term that penalizes it directly, and they show experimentally that this term improves the performance of hEP significantly in asymmetrically-initialized networks, in certain visual tasks using a 4-layer convolutional network.", "questions": "The discussion suggests that \"analog substrates could achieve this relaxation “for free” through device physics (Yi et al., 2023; Kendall et al., 2020)\". Could the authors please clarify, does any analog substrate achieve the relaxation \"for free\", or what are the requirements, more specifically? For example, the Kendall et al. reference seems to rely on a very peculiar and ad hoc hardware implementation.\n\nAlso, in the most efficient hypothetical hardware implementation, I suspect that \"for free\" is far from the truth, as, even there, the relaxation phase would include multiple weight updates, which consume power; in fact commonly more power than weight reads. Wouldn't alternative algorithms that do not have this relaxation phase be significantly more efficient? The provided references do not seem to disagree with this, so in which sense do the references support the \"free\" claim?\n\nI would appreciate the authors' insight in these questions.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "The paper is mostly nicely written and clear. The theoretical contributions of the paper are not trivial, as they require a deep understanding of a very specific algorithm i.e. the holomorphic version of EP, as well as a degree of comfort with certain mathematical concepts that is rare among neural network practitioners and possibly even theoreticians. More generally, the paper aims to contribute to an area that is of broad interest, as it relates to machine learning, neuromorphic engineering, and theoretical neuroscience. Furthermore, it truly advances the empirical results of the EP-related literature.", "weaknesses": "While this is obviously a valuable piece of work in certain respects, I believe it is also significantly limited in other key aspects.\n\nSignificance: The work ultimately aspires to improve aspects of backpropagation that are indeed important and relevant to multiple large disciplines (ML, neuromorphic hardware, neuroscience), however, concretely, the resulting contribution is very narrow. Namely, it improves the performance and the theoretical understanding of hEP somewhat, specifically in the case of asymmetric weights, but it doesn't resolve the problem of asymmetry completely, as can be seen in the performance comparison to symmetric weights (Table 2). Moreover, hEP is a specialized version of EP that makes additional assumptions for complex-valued networks, which limits the applicability and generality of the algorithm. Furthermore, EP itself more broadly is interesting but is a rather limited method in terms of achieving its goals of good performance, efficiency, useful hardware demonstrations, or deep learning, in comparison with other methods that have similar goals that have been more successful. Its biological compatibility could also be debated, given its requirement for multiple network-wide iterations before a weight update. Therefore, it could be argued that the significance of the results only relates to a very narrow subfield. Furthermore, the theoretical advance separating the impact of the finite nudge from that of the asymmetry in hEP is not trivial, but I wonder how useful it is to the ICLR community, beyond the very narrow sub-community that specializes in hEP. In the broader picture, and given the already existing better-working alternatives, the progress made here towards biological plausibility or neuromorphic computation seems small, in my view.\n\nNovelty: The main novelty of the paper is in the theoretical results, since the empirical fact that improving performance by dealing with asymmetric weights is not a new result. Nevertheless, I suspect that to someone with a good understanding of the earlier mathematical work that introduced hEP (Laborieux & Zenke, NeurIPS 2022) the new theoretical results might not be very surprising. However, it should be noted that I am not in a position to judge this fully. In any case, the insight from the theory is the impact of asymmetrical weights, relative to that of finite nudge, which seems to be also covered by the empirical result, therefore the added value from the theory in this case might not be substantial. Regarding the paper's empirical or practical contributions, some novelty exists in the objective function that penalizes asymmetries in the weight matrix. However, the novelty of this is limited because much older learning rules that achieve weight symmetry do exist (Kolen & Polak, IJCNN 1994; see also Payeur et al., Nat. Neurosci. 2021). In fact, these rules do not rely on global iterative equilibria, so the present paper's implementation of learnable weight symmetry could be characterized as a step-back in this regard. It should be recognize though that the new method is applicable when the connectivity is not reciprocal, whereas previous ones probably were not.\n\nContextualization in the literature: The paper does (now) cite some of the works that had similar aims, but only in passing, only in the discussion as opposed to the motivation section, discounts the better empirical results that the other works achieved, attributing this only to the lower simulation cost of the alternatives, and does not mention that some of these methods not only perform better in classification benchmarks, but also require fewer assumptions for compatibility with biology and for efficiency in learning hardware, e.g. by circumventing the need for backward passes of information completely.\nSome examples are: Payeur et al., Nat. Neurosci. 2021; Greedy et al., NeurIPS 2022; Mengye Ren et al., ICLR 2023; Journé et al., ICLR 2023.\n\nIt would be very helpful to the readers and the targeted research communities if the authors motivated their choice to focus specifically on EP as opposed to alternative methods that have similar goals, but don't have the same limitations. This could be a way to mitigate the weaknesses of the paper to some extent.\n\nTo be clear, some of the algorithm's limitations in comparison to alternatives are: reliance on complex-valued networks; constraints in network architecture and depth, expensive simulations for the equilibrium dynamics (e.g. a multi-GPU cluster was used according to the supplementary material, despite the small networks and simple tasks); questionable bio-plausibility of the necessity for equilibrium; hardware implementations of EP are mostly theoretical. To their credit, the authors have mentioned some of these limitations in the discussion. They also provided some solutions or counterarguments, however these are largely theoretical, vague, or speculative.\n\nAll in all, I believe that the work's value might be able to increase if the manuscript could explain its motivations and its contributions in a context broader than the EP literature. At present, this is not clear enough or supported well enough, in my view.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "Cxi3iZGbVq", "reviewer": "Reviewer_nv6j", "summary": "This paper studied the weight asymmetry in the weight transport problem extending the Holomorphic Equilibrium Propagation algorithm (hEP), through investigations about the Jacobian of the network, with experiments and theoretically analysed the reasons for the experimental results. Based on the analysis results, some new features were proposed to enable the algorithm to evade the need for perfect weight symmetry without affecting the functional symmetry. A new form of Jacobian homeostasis was introduced to maintain functional symmetry, without directly addressing weight symmetry. Finally, several experiments, including the investigation of weight symmetry evolution during training and comparative experiments, were conducted to verify the effectiveness of the proposed algorithm, and the work’s performance exceeded the networks with Recurrent Backpropagation (RBP), even on larger datasets.", "questions": "Please refer to the above weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "*Relatively new perspective\n\nThis paper shows a new perspective on studying the weight transport problem with the Jacobian of the network. It explains the connections between weight symmetry, functional symmetry and Jacobian homeostasis.\n\n*Relatively rigorous analysis and persuasive experiments\n\nThis paper shows the efforts in investigating the weight symmetry evolution during training in section 4. If the observations could be discussed more deeply with the figure would be better. \n\n*The part about ‘Jacobian homeostasis improves functional symmetry’ in section 4 is very detailed and well analysed.", "weaknesses": "*Some expressions with flaws\n\nAlthough ‘hEP’ is the abbreviation of ‘holomorphic Equilibrium propagation’ can be understood after reading. However, this abbreviation has not been indicated in parenthesis when mentioning its full term for the first time.\nThe last sentence ‘It is worth nothing that…’, in Definition 1 of section 2.2, might be miswritten, which should be ‘It is worth noting that…’ based on the context. Also in this paragraph, the sentence after Eq. (6) ‘Importantly, the quantities …, which applies only to EBMs.’, is a little longer and complex. It might confuse readers, for convenience of understanding, it would be better to split it into two simpler sentences.\n\n*Some disadvantages in the layout\n\nIn section 3.2, the paragraph with Eq. (9) is intersected by Figure 2, which could be rearranged to provide enhanced readability. The same disadvantages happen in Figure 3 and Table 1.\n\n*The performance of the proposed method is relatively not so advanced. And the method for comparison is relatively not so new, making the work of this article not very convincing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "UG8gXbu1KG", "reviewer": "Reviewer_2L2m", "summary": "The authors suggest a modification to the holomorphic equilibrium propagation algorithm, that helps it deal with cases where the Jacobian is not symmetric, by adding a term which penalizes asymmetry of the Jacobian. They show this approach outperforms standard holomorphic equilibrium propagation.", "questions": "Most important for me is to explain how this could be helpful in artificial learning systems, or why the problem of settling time does not apply to biological systems (which seems unlikely to be honest).", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The method improves over previous holomorphic equilibrium propagation in the applications tested. \n\nAs someone who is not in this field, I found section 2 to be informative introduction to the area.", "weaknesses": "It was not entirely clear to me what the motivation for this work is.  While the authors suggest that hEP is potentially biologically plausible, this seems like a stretch.  Not only does it require neurons to do computations with complex numbers, it also seems to require that the network settle to equilibrium at multiple phases of an ongoing oscillation.  This does not seem likely in the brain: the period of the gamma oscillation is tens of milliseconds, which certainly would not allow enough time, and even the theta oscillation seems to fast for this. It also was not clear what computational advantages this might lead to in purely artificial systems for which biological plausibility was not important.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)