Curl Descent : Non-Gradient Learning Dynamics with Sign-Diverse Plasticity
Authors: Hugo Ninou
Jonathan Kadmon
N Alex Cayco Gajic
Venue: NeurIPS 2025 spotlight ()
Keywords: learning dynamics
non-gradient
learning rules
Hebbian
plasticity
anti-Hebbian
Excitatory-inhibitory
biologically plausible
biological networks
theoretical neuroscience
teacher-student
non-reciprocal
Primary area: theory
TL;DR: In sign-diverse Hebbian/anti-Hebbian or E-I networks, inherent non-gradient “curl” terms arise, and can, depending on network architecture, destabilize gradient-descent solutions or paradoxically accelerate learning beyond pure gradient flow.

Abstract: Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient-based strategies during learning. Experiments often discover a diversity of synaptic plasticity rules, but whether these amount to an approximation to gradient descent is unclear. Here we investigate a previously overlooked possibility: that learning dynamics may include fundamentally non-gradient "curl"-like components while still being able to effectively optimize a loss function. Curl terms naturally emerge in networks with excitatory-inhibitory connectivity or Hebbian/anti-Hebbian plasticity, resulting in learning dynamics that cannot be framed as gradient descent on any objective. To investigate the impact of these curl terms, we analyze feedforward networks within an analytically tractable student-teacher framework, systematically introducing non-gradient dynamics through rule-flipped neurons. Small curl terms preserve the stability of the original solution manifold, resulting in learning dynamics similar to gradient descent. Beyond a critical value, strong curl terms destabilize the solution manifold. Depending on the network architecture, this loss of stability can lead to chaotic learning dynamics that destroy performance. In other cases, the curl terms can counterintuitively speed up learning compared to gradient descent by allowing the weight dynamics to escape saddles by temporarily ascending the loss. Our results identify specific architectures capable of supporting robust learning via diverse learning rules, providing an important counterpoint to normative theories of gradient-based learning in neural networks.

Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity
Hugo Ninou Département d d'Études Cognitives École normale supérieure- PSL hugo. wwwonnenn......e fr
Jonathan Kadmon
Edmond and Lily Safra Center for Brain Sciences
The Hebrew University of Jerusalem
j jonathan kadmon@mail hujji. mujj.aa ac. il
N. Alex Cayco-Gajic Département d' E'Euddes Cognitives École normale supérieure- - PSL natasha. cayco gajiceens. fr
Abstract
Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient- based strategies during learning. Experiments often discover a diversity of synaptic plasticity rules, but whether these amount to approximation to gradient descent is unclear. Here we investigate previously overlooked possibility: that learning dynamics may include fundamentally non-gradient "curl"-like components while still being able to effectively optimize loss function. Curl terms naturally emerge in networks with inbibitory-exciiatory connectivity ticity, resulting in learning dynamics that cannot be framed as gradient descent on any objective. To investigate the impact of these curl terms, analyze feedforward networks within an analytically tractable student-teacher framework, systemati- cally introducing non-gradient dynamics through neurons exhibiting rule-flipped plasticity. Small curl terms preserve the stability of the original solution manifold, resulting in learning dynamics similar gradient descent. Beyond critical value, strong curl terms destabilize the solution manifold. Depending on the network architecture, this loss of stability can lead chaotic learning dynamics that destroy performance. In other cases, the curl terms can counterintuitively speed learning compared gradient descent by allowing the weight dynamics to escape saddles by temporarily ascending the loss. Our results identify specific architectures capable of supporting robust learning via diverse learning rules, providing an important counterpoint to normative theories of gradient-based learning in neural networks.
hathaanntt-Hbaannn plas-
1 Introduction
Modern deep learning relies on backpropagation to compute nigh-dimensional gradients that assign credit individual synapses by propagating error signals backward through the network [Rumelhart jet al. 1986 Chinta and Tweed 2012). This mechanism solves the credit assignment problem by assuming that each synapse can locally adapt in proportion the negative gradient of the loss. However, despite its central role in artificial systems, direct evidence for gradient-based learning in biological neural circuits is still lacking [Lillicrap
2020
To reconcile this gap. many studies have proposed biologically plausible approximations to gradient descent [Richards and Kording 2023). These typically address concerns related to the locality of
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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error information [Golkar et al. 2023 Keller and Mrsic-Flogel 2018 Bredenberg et al., 2023].
separation of forward and backward passes Guerguiev et al. 2017 Xie and Seung, 2003 and the
weight transport problem [Akrout et al. 2019 Lillicrap et al. [016) However, more fundamental constraint has received less attention: even if local gradient information were available, remains unclear whether biological plasticity rules can consistently drive synapses along such gradient.
Backpropagation implicitly assumes a coordinated update rule in which all synapses adjust their weights in directions aligned with the descending gradient of a global error signal. Yet this assumption incompatible with experimental observations. Synaptic plasticity in the brain is remarkably diverse, with different cell types and circuits expressing distinct, and sometimes opposing, forms of long-temm -teet potentiation and depression, including both Hebbian and mnti-Hebbian rules [Abbott and Nelson 2000]. This diversity is compounded by Dale's 's law (Dale 1934], which fixes each neuron as either excitatory or inhibitory, constraining the sign of its outgoing synapses. Crucially, the local plasticity rule at given synapse appears uncorrelated with the identity of the presynaptic neuron wehther excitatory or inhibitory- or with any other structural or physiological feature of the circuit (Citri) and Malenka 2008). As a result, identical local signals can produce opposite weight changes across different synapses, with no apparent mechanism to coordinate or compensate for this variability, These constraints raise more fundamental question: can networks with heterogeneous and potentially antagonistic plasticity rules still support meaningful optimization?
In this work, we examine how physiological diversity in synaptic plasticity and cell-type identity, shapes the learning dynamics of neural networks. Specifically, we ask whether networks composed of heterogeneous neurons can still effectively reduce an objective function or whether such diversity precludes gradient- based learning altogether.
Our contributions are as follows:
We show that non-gradient terms naturally arise in biologically plausible networks due to sign-diverse plasticity. In contrast to many previously considered alternatives to backpropa- gation, here the learning dynamics cannot be written as gradient descent on any objective due to the existence of non-gradient "curl"-like terms.
We develop a theoretical framework to isolate and systematically analyze the effect of curl terms in large linear feedforward networks. Leveraging random matrix theory, we identify dynamical phase transition which the zero-error solution manifold loses stability.
We demonstrate that the location of this phase transition depends on architectural parameters, particularly the expansion ratio between the input and hidden layers.
Finally, we provide numerical evidence that, certain nonlinear architectures, curl descent can accelerate learning, even in the absence of true gradient flow.
Our results suggest a previously unexplored mechanism through which biological learning rules could give rise to fundamentally non-gradient dynamics that still support effective learning.
2 Non-gradient terms arise in biologically plausible neural networks
Our curl descent learning rule introduces non- con-gradient terms into the learning dynamics by flipping the sign of the gradient descent update for selected synapses. To motivate this approach, in this section we demonstrate how non- con-gradient learning dynamics can arise from sign-diverse plasticity in biological networks. We return to the curl descent rule in the following section.
Unlike artificial networks, neurons in the brain exhibit variety of plasticity mechanisms and physiological properties that directly influence how a synapse is updated in relation to the local gradient. We begin our analysis by showing that sign diversity in effective learning rules whether from plasticity mechanisms or the neural dynamics of excitatory-inbibitor networks - gives rise to provably non- non-gradie -nngrradient terms in the learning dynamics. While we focus on recurrent linear architectures for brevity, it is straightforward to extend this analysis to nonlinear dynamics.
Excitatory-nnhibitory (E-I) networks. Non-gradient terms can emerge in recurrent B-I Hebbian networks. Consider a linear recurrent neural network (RNN) described by:
TW yy y+ + WD +f, for D = diag(d)........ dyy with d, (+1,-1)
(L)

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NxN denotes non -on-nngative where E RN represents the firing rates of the N neurons, W E 22 recurrent weight magnitudes, d, determines whether neuron is excitatory or inhibitory, Ty > 0 is a time constant, and f E Rº is an external drive. The Taylor expansion of the neural dynamics at steady state gives y (II + WD + + Under Hebbian plasticity (with weight decay), obtain the following learning dynamics:
Tw-W=y>
> R ff' #WDf ff' -WW
Here, Y > 0 regulates weight decay prevent unbounded growth [Gerstner and Kistler 2002] While the first and last terms can be written as gradients - namely, ff Vw Tr (ff )
and >W = 2/2 Vw Tr (WWT)- - the two intermediate terms cannot, unless ff' D is symmetric. However, this symmetry holds only when the inhibitory neurons receive external input; Supplementary Material A.3 for details. This is the case in previous normative studies that derive excilatory-inhibitory Hebbian networks from similarity matching objective [Pehlevan et al. 2015 (see Supplementary A.44). In the general case, networks that respect Dale's s law will therefore include non- con-gradient terms in their learning dynamics.
Hebbian/anti-Hehhee networks. A similar argument can be made for networks where the learning rule of individual synapses are sign-flipped as result of mixed Hebbian and anti-Hebbian plasticity; see Supplemenaary Material A.22 for details.
3 Curl descent in a student-teacher framework
To better understand the effect of non-gradient terms (here called "curl" terms in analogy with the
Helmholtz decomposition), dynamics of gradient descent is well understood [Saxe et al. [2014] Advani et al. 2020 Goldt et al.
next turn to an analytically tractable setting which the learning
2020, Baldi and Hornik 1989, Le Cun al. 1991 Seung et 1992 linear feedforward networks with single hidden layer.
We adopt a student-teacher framework in which two-layer teacher network (parametrized by weights W E NM and W3 E (R-V) maps an input vector x E R& to scalar output y E R via y W/ x. The student uses the same architecture, and its output is given by ý = W2h, where h = Wix is the hidden layer activity. The student's ' goal is to modify its weights W1 and W2 to match the teacher's output y and minimize the quadratic loss C - (e²), where e ÿ y the signed error and (.) denotes an average over the input distribution.
Standard gradient descent gives the following updates:
A AW = Ww, C = W. W2Wxxx - W. yx = W ex
(3)
A AWardd Ww,C - Wihh tyh -ch
(4)
Each term is the outer product of a postsynaptic error signal and the presynaptic activity, and can be considered a supervised "Hebbian-libe like"" learning rule [Melchior al. 2024 Refinetti et al. 2021
Curl descent rule. To model the diverse behavior of plasticity rules observed in biological neural networks. flip the sign of subset of synapses using diagonal matrices D, E IR MRM and D2 E NxN;
AW = - W. ex D1,
AW = -ch
(5)
where D, = diag(di,,, (di, di,M). D2 = diag(d2, (2,1,, d2.)), and dr,; E (-1,-11. Therefore, all synapses associated with presynaptic neuron in layer I follow either unchanged learning rule (di, = +1) or flipped learning rule (dr,, = - If both types of learning rule are present in the student, no scalar potential function exists whose gradient reproduces the weight updates (see Supplementary Material). Thus, the addition of rule-flipped plasticity induces intrinsically non-gradient curl terms in the learning dynamics.
4 Analytical results
Following previous work on the learning dynamics of linear networks [axx et al. 2014), we assume whitened inputs (xxT) Ia and take the continuous time limit (small learning rate) giving the

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following nonlinear dynamical system for the weights:
W W (s D2 and
D2.
(6)
where = W. W represents the effective function implemented by the teacher. If D1 LM and D2 = IN. the learning dynamics reduce to gradient flow.
Since curl descent only changes the sign of the plasticity rule, it will have the same fixed-point solutions as gradient descent: these include a continuous manifold corresponding to W,W; (here called the solution manifold) plus discrete fixed point at the origin (W₂ W2 = 0). In gradient descent, the hyperbolic solution manifold is known be stable, whereas the origin is saddle (Saxe et al. [0144). However, curl terms can have significant impact the learning dynamics by changing the stability properties of fixed points. Flipping the sign of all the synapses would have clearly disastrous impact it would lead the weights to ascend the gradient. Surprisingly, however, the stability of the solution manifold can be robust to moderate amounts of rulc-ffipped plasticity depending the network architecture, as we will demonstrate below.
4.1 Toy example: A two-neuron network
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Toy model analysis. a) Learning trajectories in weight space for gradient flow (dark purple curve) and curl flow (light purple curve). The heatmap represents the log loss, which determines the gradient descent dynamics. The hyperbolic solution manifold (dark red curves) is global minimum. Curl descent reshapes the learning dynamics and adds a rotational field (flow-field overlain in light purple curves with arrows). b) Schematic of the toy model network. e) Log error vs. training epoch for the same learning trajectories shown in panel a. Inset: Same figure zoomed on the first 10 epochs, showing that curl descent initially ascends the loss function.
Figure I:
We first build intuition by considering minimal wwo-neuron network (M N 1). We will denote the resulting scalar weights of the teacher and student w/ and wi, for I 1, The continuous-time gradient flow dynamics are given by:
w Wa (s w2w1) ),
úh = wi (s w2w1)
Flipping the sign the hidden neuron's plasticity rule instead yields the following dynamies:
w w2( w2w1),
t wi wyw1)
How does this sign flip modify the stability of the solution manifold? To test this analyze the change of stability of fixed points by calculating the curl flow Jacobian:
=
(s 2w2wi) )
(9)
(2w2w1
wf
On the solution manifold (s w2)), the eigenvalues are X = O and X2 w w Hence, only the fixed points satisfying |wa| > |wn| will remain (neutrally) stable, while the other half of the solution manifold loses stability.

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The origin, saddle under the original gradient flow dynamics, is converted to center with purely imaginary eigenvalues 21 = -is (Fig. Ia). The phase plane shows two qualitatively distinct dynam- ical regimes: convergence to minimum on the solution manifold or small-an small-amplitude oscillations, depending on the initialization of the weights. These regimes are separated by heteroclinic orbits on the circle wf + wg = 8.
Under curl flow, the weights evolve according rotational dynamics induced by the curl terms, but can still descend the loss and converge to the solution manifold. Intriguingly, in some cases curl flow can lead to faster convergence compared gradient flow (Fig. 1). In particular, this happens when the weights would otherwise be stuck along the stable manifold of the saddle at the origin. This hints at possible benefit of curl terms in helping weight dynamics escape saddles, but comes at a significant cost: half of the solution manifold has lost stability, and small- -mppliude initializations no longer converge, This can partially be explained by the fact that in this two-neuron network, we flipped the sign of an "entire layer" to add curl term. In the following section, we consider large networks where we have finer control over the magnitude of the curl terms.
4.2 Large networks
Returning to the general case (arbitrary M, N), we derive the Jacobian J/ of the learning dynamics at the origin and on the solution manifold. The eigenvalues parameter space reveal the local stability at this point: if all eigenvalues' real parts are negative, then the point is stable, otherwise it unstable. We thus analyze the eigenvalues of the Jacobian we systematically increase the fraction of rule-flipped neurons in either the hidden layer or the readout layer. For this, we quantified by the fraction of negative diagonal elements in D₃ or D2 as:
the Jacobian matrix at a given point in
M
N a, = Σ
1 M
Oh/
Σ
(d1.5 < 0},
1d2s < <0}
(10)
where denotes the indicator function (derivation details can be found in the Supplementary Material).
Spectrum at the origin.
At the origin, the characteristic polynomial can be derived as:
det(J M1) (-A)MN-N N II d2
Σ== Σ d d
(1I)
1=1
Therefore, the origin can have at most 2N nonzero eigenvalues: À A= = In the case of gradient flow, all dr,j = and Σ > 0. yielding purely real cigenvalues of positive and negative sign (a saddle). If we now add rule-flipped plasticity in the readout layer, then every hidden neuron whose plasticity is sign-flipped (i.e., for each d2,; = -1) will convert two of those eigenvalues to be purely imaginary, turning one of the N planes with embedded saddle dynamics to center point. Instead, we add sufficient rule-fling allp-fi plasticity to the hidden layer (enough so that Σ < 0), all of the saddle planes will turn into centers. This strong dependency the layer foreshadows the important role that the network architecture will play in determining how curl flow impacts learning dynamics.
Spectrum on the solution manifold. Using the Schur complement, can derive the characteristic polynomial of evaluated on the solution manifold (W, W2 = 8) as:
det([ - A/) = NM+ M det( Al + |W2| DD + W,T D2W1).
(12)
Hence, most M eigenvalues are nonzero and their values will be governed by the determinant on the right-hand side in (12). In the case of gradient flow, this reduces to the characteristic polynomial of a second matrix:
Let(Alas + A) - A [W2 + W.T W1
(13)
Since A is positive definite, the eigenvalues of A, and hence the nonzero eigenvalues of J, are negative, demonstrating that the solution manifold is stable under gradient flow.
How does the stability of the solution manifold change under curl flow? If we flip the sign of all neurons (a,, a, 1), all eigenvalues will flip their sign in correspondence. However, there may be intermediate values of on or a, before the solution manifold loses stability. Indeed, we can directly infer from the structure of Eq. (12 that stability depends on two factors: the ratio between
5

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the variances of W1 and W2 given by the ratio M/N, and the fraction of flipped synapses, either ar, or a,. depending on the layer being modified. This simplification arises because the stability is determined by the point at which the largest eigenvalue is zero and does not rely on other properties of the eigenvalue distribution.
The characteristic polynomial is difficult to evaluate in general, but we can leverage random matrix theory to predict when the bounded support of the eigenvalue distribution crosses zero in large networks. Here we use the i.i.d. random distribution of teacher weights, and assume that the student weights share the same statistics on the solution manifold (He et al. 2015], namely:
(Wi)a lid V(0,1/M) and (W2),; lid N/(0
(14)
We consider the infinitely wide limit (M, N 20) with fixed "compression" ratio c. M/N, allowing us to characterize how the stability properties change as a function of the network architec- ture. First, we ask how the spectrum of the Jacobian changes as we vary an (keeping a, = 0). In this case, we can derive fourth-order polynomial whose double roots provide the endpoints of the spectral support (see Supplementary Material). The double roots can be solved numerically obtain the stability boundary as a function of e (Fig. 2 top). The stability boundaries for the complementary case, in which we instead vary a, (keeping 0h 0), can be found using the method of [Kumar and
Sai Charan 2020 (Fig. 2 bottom).
Curl-induced destabilization depends network architecture. The phase diagrams in Figure 2 highlight a clear trend: expansive networks (c < 1) are inherently more robust to curl terms. When added to the hidden layer, curl flow destabilizes the solution manifold once the compression ratio exceeds e 0.3 (Fig. 2 top). This critical value depends only weakly on ag. In contrast, for the readout layer (Fig. 2 bottom), the stability of the solution manifold depends
Wi
N
W.
y
Hidden layer
Solution mariald statle
strongly on the magnitude of the curl terms. In particular, at most half of the readout layer can ohey rule-flipped plasticity before stability is lost. These results demonstrate the conditions under which curl descent may converge to the same solution manifold as gradient descent. To understand how the learning dynamics change at the stability boundary, we turn to simulations.
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To test our theoretical results on the stability of the solution mani- fold, we simulated networks with total of M N -220 neurons, while varying the compression ratio e and the fraction of rule-flippe file-flipped neurons (either a, or (n)) The hidden and read-out weights of the teacher were sampled i.i.d. from pero-mean distributions, with vari-
0.6
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ance scaled by the number of input neurons to each layer, ensuring that stability depends only on the compression ratio M/N and
10
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not on the statistics of the weights. The student networks had iden- tical architectures to the teacher networks, with weights initialized from the same distribution (unless otherwise specified). Inputs were sampled as x; ",;'((O 1/v2), and along with the teacher's outputs, provided the training data for the students. Weight updates were made on the whole training set (Nimin = 250 samples) with a learn- ing rate 0.1/Narin over Nopochs = 100h epochs. To ensure numerical stability, W1 and W2 were re-normalized at every epoch to match their initial Frobenius norm. When analyzing the stability regimes, we focused on linear networks to be able to compare directly to theory; however, we note that the qualitative properties also extend to nonlinear networks (see Supplementary Material C Furthermore, in final results convergence speed in curl descent we implemented nonlinear networks with tanh activation functions.
Figure 2: Analytical phase diagrams. Stability of the solution manifold as a func- tion of the compression ratio and the fraction of rule-flipped neurons in each layer as (hid- den) and at (readout).
Dynamics beyond stability. In our analytical results, we have shown that above the critical values of e and o6 or Or,, the solution manifold loses its stability. What happens to the weight dynamics in this case? Interestingly, this depends on which layer we are flipping: hidden readout.

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Test error
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Figure 3: Hidden layer curl terms lead to chaos. a) Test error as a function of the compression ratio e and the fraction ule-ffipped neurons an (averaged over 10 random seeds). Black curve: analytical stability boundary (ef. Fig. 2 top). Inset: Close-up for E [0.1, 0.7]. b) Order parameter (averaged over 10 seeds) plotted for varying ar (top) and (bottom). Dashed lines indicate analytical transition to instability. e) Example weight dynamics in the unstable regime (c = 0.8, at 0.6). d) Example weight autocorrelation functions. Inset: Weight dynamics projected onto its first two principal components. Compute resources: 4 hours on 500 CPUs (local cluster).
Rule-flipped plasticity the hidden layer. Our simulations show that introducing rule-flipped weights in the hidden layer induces chaotic learning dynamics when the solution manifold loses stability (Fig. 3 The emergence of chaotic weight dynamics is analogous to the transition to chaos in large disordered networks (Kadmon and Sompolinsky 2015]. This occur when the Jacobian always has at least one unstable direction. Notably, our analysis found this to be the case on the solution manifold. Still, simulations suggest that the dynamics everywhere unstable, be seen from the order parameter quantifying the mean fluctuations:
E ((w (w)) [(())
with w == (Wi)iy for the sake of notation
(15)
where (-) represents average over epochs, and E|-}, V represent mean, variance over (.j. Here, the transition to chaos appears even in linear networks, because the learning dynamics themselves are inherently nonlinear [Saxe et al. 2014). The resulting chaos can be understood as a result of nonlinear weight update combined with structural (quenched) disorder (Sompolinsky et al. 1988).
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Figure 4: Readout layer curl terms result in low error even when the solution manifold is unstable. Low test error with readout curl terms. Same as Fig. 3a while varying o,- The black curve shows the analytical stability boundary. b) Peak error over learning (maximum over 20 random seeds, initialized near the solution manifold). Inset: Test error vs. epoch the unstable regime, showing large weight transients that re-descend the loss (c = 1, a, = 0.6). e) Example weight dynamics in the unstable regime (c 1, 0,, 0.6). Example weight autocorrelation functions. Inset: Weight dynamics projected onto its first two principal components. Compute resources: hours on 500 CPUs (local cluster).
Rule-flipped plasticity in the readout layer. Surprisingly, destabilizing the solution manifold by introducing rule-flipped plasticity in the readout layer did not necessarily prevent the network from reaching small test error (Fig. 4... To verify that the solution manifold was indeed unstable, tried initializing the student networks' weights away from the solution manifold by adding a 10- perturbation. The typical eтror evolution showed a spike the loss before going down another stable minimum (Fig. 4Dbdd), reminiscent of the dynamics of the two-neuron model in Fig. [ The dynamics suggest that solution manifold was indeed unstable, but the weights were nevertheless able to find other low-error regions of the parameter space. We suspect this difference, compared with the chaotic learning dynamics observed when including rule-flipped plasticity in the hidden layer, may be due to the dinennmensional (scalar) output.

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Figure 5: Nonlinear networks: curl descent leads to faster convergence in a broad parameter regime. a) Network schematic. b) Test error for gradient descent and curl descent with single rule-flipped readout neuron (NVrain = 2000, weight initialization scale = 2: error bars indicate mean + sem, averaged over 10 random seeds). Inset: activation function (tanh). e) Convergence speed of curl descent and gradient descent as a function training set size (weight initialization scale = 2). function of the weight initialization range (Nurain 10000). Convergence speed
d) Same as a function the teacher weights initialization scale. f) Convergence speed as a function of the fraction of rule-flipped readout neurons. Compute resources: 12 hours on 500 CPUs (local cluster).
Faster convergence in nonlinear networks. These results led us to ask whether curl descent could have any numerical advantage compared gradient descent. In the toy example, we saw that in some circumstances, curl descent could descend the loss faster than gradient descent (Fig. []. To test this, we simulated contracting tanh networks with M 100 input units and N 10 hidden units while flipping the learning-rule sign for single weight in the readout layer: i.e.. only of the model's 1010 trainable parameters (Fig. ITp.. Indeed, a single flipped weight was able to significantly reshape the learning trajectory, leading to faster convergence (Fig. Tp). This improved performance
increased in the high data regime (Fig. ITc. < .01 for Niain 2 500, paired t-test) and was robust to broad range of student weight initialization scales (Fig. l1d), H), where Wrstddet scale Wsstudlett
We next tested whether convergence speed differences generalized to more complex tasks. Using prior results [Poole et al. 2016 Bahri et al. 2020), we modulated the teacher's function complexity by parametrically expanding the initialization range of the teacher weights. We observed a critical value of the teacher weight initialization scale (Ti) above it both curl descent and gradient descent exhibited sharp reduction in performance with both rules performing similarly poorly, but below it, curl descent consistently outperformed gradient descent (p < .01, paired t-test).
Lastly, investigated the robustness of curl descent's 's rapid convergence by increasing the proportion of rule-flipped neurons in the readout layer (Fig. ITp. As expected, the speed advantage of curl descent diminished as the fraction of flle-fliped neurons rose, as the resulting learning dynamics became increasingly unstable. However, curl descent was nevertheless able to descend the error faster than gradient descent for 50% rule-fipped readout weights (p < .01, paired t-test).
Further tests will be necessary to fully resolve the impact curl descent in complex, real-world tasks. Nevertheless, numerical results demonstrate that in broad range of hyperparameter settings, curl descent can counterintuitively speed learning by allowing the weight dynamics to find other low-error solutions than those found by gradient descent.
Related work
Non-equilibrium neural dynamies. The curl descent rule induces non-equilibrium learning dy- namics by breaking the symmetry of the Jacobian. Such non-gradient systems have also been

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studied in the context of complex networks in the absence of plasticity, including in brain dynam- ics [Nartallo- Kaluarachchi et al. 2025 Daie et al., 2025]. Here, non-reciproeal connections between neurons introduce curl terms that can give rise to a host of different dynamical regimes (Yan et al. 2013 Fruchart et al., (021]. To our knowledge, this the first study to systematically investigate non-equilibrium learning dynamics.
Natural gradients. Natural gradient descent |maai (998) replaces the standard gradient descent
update with AO - G- VoL, where the preconditioning matrix G is positive definite. Selecting G equips the parameter space with a Riemannian geometry, determining the learning flow field (Surace) ct al. 2020]. When G is positive-definite, each step is guaranteed to decrease the objective, yielding monotonic improvement [Richards and Kording 22233 Richards et al., 2019]. Non-Euclidean metrics have been shown to better reproduce observed weight distributions in the brain (Pogodin et al. 2023] Curl descent does not induce a Riemannian metric: the corresponding preconditioning matrix is indefinite, possessing both positive and negative eigenvalues. This violates the assumption of natural gradients. producing weight dynamics with qualitatively new behavior, including rotational vector fields and periodic orbits the parameter space.
These new dynamics are interesting to compare to recent work which argues that a wide class of learning rules can be viewed as natural gradients for particular metric [Shoji et al., 2024). The authors give a constructive proof that any rule which monotonically decreases cost function over some time window implements form of natural gradients. However, this condition is not satisfied by curl descent: for example, in the toy model in Fig. la, small amplitude initial weights result in periodic dynamies. In simulations in larger networks, we did not observe nenrescegg loss curves (as long as only a minority of neurons in the readout layer were flipped), but we cannot a priori rule out the existence of pathological weight initialization regimes that would lead such behavior.
Feedback alignment. A growing body of work uses random projections of the error a biologically
plausible mechanism to circumvent the weight transport problem [Lillicrap et al. [016)] Nok
2016 Refinetti et al. [02T Hanut and Kadmon 2025 Clark et al.) [221] Moskovitz al. 2019
Lindsey and Litwin- Kumar 2020. Boopathy and Fiete 2022]. The weight updates are then given by: AW,- Bex and AW2 -eh". where B is feedback matrix. If B W.T. we recover gradient descent. Unless BW2 is symmetric, this learning rule cannot be expressed as deriving from a gradient. In classic feedback aligmment, B is chosen to be random, therefore, this condition is unlikely to be satisfied (note that W2 has been numerically observed to partially align to B through the weight dynamics (Lillicrap et al. 2020)). The fact that this algorithm offers little control over the non- -on-grddient terms makes it difficutt to test their effect systematically. Curl descent enables this control by flipping the learning rule for randomly selected fraction of neurons.
Exact learning dynamics. Our work extends previous analytical studies of exact learning dynamics of gradient descent in linear neural networks [Saxe et al. 2014,] Advani et al.) 20200 Pellegrino et al., 2023, Bordelon and Pehlevan| 2025 Hanut and Kadmon 2025 A contribution of our framework is to devise a tractable learning rule that enables mathematical analysis of how different amounts of non-gradient terms influence weight dynamics. Unlike previous work, our model highlights cases where the dynamics cannot be captured by a potential.
Normative theories of Hebbian learning. Decades of work has shown that Hebbian-like learning rules can optimize specified objectives (Bahroun et al. [023 Melchior et al. 2024 Pehlevan et al.
2015] Pehlevan and Chklovskii 2019
Tolmachev and Manton 2020 Hyvärinen and Ojal 1998
Seung and Zung [017 Foldiak
1990
Seung 2018 Xie and Seung 2003 Lim 2021 Obeid et al.
2019 Halvaga and Zenke 2023 Flesch et al. 2023 Brito and Gerstner 2016 Lipshutz et al. 2023
O O'Reily 2001 Eckmann et al.
2024 [Pehlevan et al. 2015] propose a biologically plausible neural network with Hebbian updates for the excitatory feedforward connections and rule-flipped updates for the lateral inhibitory neurons. The design of this specific network architecture effectively annihilates the curl terms, enabling their Hebbian/anti-HH Hehbiin learning rules to optimize a similarity matching function. Our work demonstrates that in more general architectures, the sign diversity of Hebbian/anti-Hobbal learning rules induces curl terms into the dynamics.
Excitatory-Inhibice networks. Recent work derived learning rules for Dale's law-compliant net- works from optimization principles [Cornford et al. 2024]. Other studies have found that excitatory-

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inhibitory plasticity enhances memory formation and retrieval in neural networks [Gong and Brunel 2024, Vogels et al., 2011 Michl and Gjorgjieva 2022 Wu al.) 2022 Agnes and Vogels| [024]. without explicitly deriving these rules from a cost function. Our work argues that curl terms originat- ing from the sign diversity inherent in excitatory-nnhibioryy networks could contribute to improved task performance, using mechanism distinct from the standard optimization view. In addition, it proposes constraints on the architectures that can support gradient learning with inhibitory neurons.
Discussion
Here, we demonstrated that non-gradient terms arise due to sign diversity in biologically motivated plasticity rules. We further developed a controlled framework to quantify the impact of increasing non-gradient "curl" components in neural learning. Our results show that depending on network architecture, curl terms can generate chaotic learning dynamics, or they can counterintuitively descend loss function even if the gradient descent solution manifold is no longer stable in some cases, even converging faster than gradient descent. More broadly, we have argued for a need to investigate how non-gradde non-gradient learning dynamics may play a role in task performance in neural networks.
We have shown how casily curl terms could arise through sign-diverse cell types (i.e., E-I) or through sign-diverse plasticity rules. However, gradient descent and sign diversity are not inherently incompatible. For instance, cell-specific plasticity rules might simply reflect how gradient- based optimization manifests differently across neurons, compensating for variations in intrinsic properties or connectivity motifs. For example, cell-specific plasticity in E-I networks can implement gradient descent of similarity matching objective (Pehlevan et al./ 2015). In Supplementary Material |A.4 we show that such networks are able to implement gradient flow by choosing a specific architecture that nullifies curl terms. Sign diversity has also been reported within single neurons, depending on the distance of the synapse to the soma [Froemke) 2010 Sjöström and Häusser [006]: this effect has been argued to be compatible with specific implementations of gradient descent (Richards et al.) sign-flips can arise as natural consequence of negative
2019]. Similarly, caaromoddulatorinndueee reinforcement in modulaadd plasticity rules [Frémaux and Gerstner 2015].
Another possibility that different plasticity rules could represent competition between distinct objec- tives locally optimized by different neural populations, such as error minimization versus homeostatic regulation. Indeed, previous work has argued that an interplay of diverse plasticity rules may underlie stable network performance [Confavreux et al., 2025 Zenke al., 2015). In the curl descent we examine, however, the plasticity rules do not differ in functional form but instead exhibit opposite signs. Each rule-flipped neuron effectively attempts to ascend the gradient, meaning its cost function becomes the negative of the original loss. These neurons can thus be viewed not merely as competitive but as adversarial. From this perspective, is particularly striking that the learning dynamics remain robust to such adversarial neurons - and, in some cases, are even accelerated by their presence.
Our results rest on several assumptions that could be relaxed in future work. First, considered only i.i.d. inputs; structured stimuli may alter stability and convergence properties. Second, our analysis focused the local stability of critical points, whereas full investigation of the accelerated convergence observed with curl descent will require a detailed treatment of the global nonlinear dynamics. Third, limited curl rules to presynaptic identity-based sign flips; exploring fine- grained, synapse-specific sign flips, perhaps correlated with the structural or neuromodulatory factors mentioned above, could reveal additional regimes. Fourth, we restricted our study two-layer feedforward networks with scalar outputs, whereas the recurrent and deeper architectures common in the brain may exhibit qualitatively different curl-induced phenomena.
Despite these limitations, our framework integrates readily with existing learning rules - gradient descent, feedback alignment, and natural- -randiet methods - by treating curl terms as a tunable perturbation. Overall, our results challenge the dominant view that effective learning must follow a gradient (Richards and Kording, 2023). Instead, biologically plausible diversity in plasticity rules may support optimization through intrinsically non-gradient mechanisms. This suggests that what may appear as biological irregularity could in fact be feature: evolutionary strategy that leverages non-gradient dynamics for efficient and robust learning. Embracing this perspective could inform new optimization principles and architectures in machine learning, expanding the landscape beyond traditional gradient descent.
Code accompanying our paper is available htttps///github cctp///aoooaaaicaaaa/tl...
10

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Acknowledgements
This work was funded by a CDSN doctoral scholarship from the ENS (HN), the Agence National de Recherche (ANR-17-726 EURE-0017, ANR-2-II---2----- INSERM, and the Gatsby Charitable Foundation (JK).
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A Learning rules that can not be expressed as gradient descent
In this section we provide proofs that certain learning rules cannot be written as the gradient of any objective.
A.1 Curl descent learning rule
We will first demonstrate that the curl descent learning rule for feedforward networks (Equation i the main text) cannot be written as gradient. Here we provide a proof by contradiction for the readout layer (AW, -eh D2). An analogous derivation for the hidden layer (AW1 = ex D₂) shows that it too cannot be obtained as the gradient of any function.
Proof. Suppose there exists energy function Len (W2) such that - ww,, = -ch D2. In that case, the ijth element of the gradient is given by:
eh D2
(16)
@W2 1g
(U vih D2
(17)
[Whhh" T D2 yh D2jes
(18)
N
W2,th hihi Dz] - Juh" D2
(19)
Since fen has continuous second derivative, we may apply Schwarz' theorem:
32 Len
8 Cell
(20)
WW2.d0W2.r
aW2
Substituting equation 19 in 20
obtain
N W2 D2lus
W2. [hh
D2las
(21)
aW
2W2.
[hh T D2lı [hh] D2l5t-
(22)
Therefore, hh T D2 must be symmetric matrix. However, since D2 is a diagonal matrix, it will rescale the äth column of hh (itself symmetric) by the D2 This can only result in a symmetric matrix if D2 + Therefore, when D2 is sign diverse, there exists no function Len for which the curl descent rule can be written as gradient descent.
A.2
networks
Consider linear recurrent neural network (RNN) in which synaptic plasticity can be either Hebbian anti-Hebbian. The RNN dynamics are given by T -y + Wy where y IR- the firing rates of the N neurons in the network, W NNN are the recurrent weights, Ty 0 is the time constant, and f IRM is external drive. We consider sign-diverse learning rule where any synapse can be either Hebbian or anti-Hebbian:
W = yy O M, with M E NxN with elements Mij E {+1,-1}
(23)
where @ denotes the Hadamard product. If synaptic changes occur on much slower timescale than neural dynamics, we can assume that the firing rates reach a steady state. Since individual weights are typically order 1 / /NN or smaller (Rubin al. 2017], the steady state can be written as y (T -f (T+ W + O(W2)) The weight dynamics are then given by
Tw W ~ T + WT + WWf O.
(24)
Here, the first term can be written as the negative gradient of an objective function: fF e M Vw Tr (-(ff) @ T) However, the last two terms cannot be generally written the gradient
25

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of any function (unless specific assumptions are made on M and W). Thus, the sign diversity of the plasticity rule results learning dynamics that are governed by both gradient and non gradient terms.
Proof. The first term can be expressed as the negative gradient ff @ M = V WWW),, where C(W)= = - Tr (fft o M) WT.. It will therefore suffice to demonstrate that the second and third terms cannot be written gradient. These terms be grouped together
F(W) = (EWT + W2) C M,
(25)
where we have defined the covariance- like matrix for the inputs as:
E = (or Σ == == (ff for time-varying inputs).
(26)
A necessary condition for a dynamical system to define gradient flow is that its Jacobian matrix is symmetric at every point W (Perko 2001). This symmetry condition implics that:
(F2(W)
(Frr(W) aw,j
Vi,j.k,
(27)
oW
In the general case that W is not symmetric, this condition reduces
(6fx Exe + 65 Στj) My = (ose Eyk + dix Σes) Mue.
(28)
In particular, setting i = = k and l + gives
Ea Mi E④ Ma.
(29)
which would force Σn = 0 for * Since rank-one, this would require no more than one neuron receives input (or, in the time-varying case, it requires inputs to be uncorrelated). If neither of these assumptions holds, there is no choice of M (which elements are 1) can symmetrize the Jacobian.
A.2.1 Special case: Gradient flow for symmetric W and homogenous plasticity
We have shown that in the general case, Hebbian/anti -ebbinn networks cannot be written as gradient descent. However, for specific choices of the architecture, the dynamics can follow a gradient. For example, suppose W W T, To ensure this holds for all time, the weight dynamics must also be symmetric, which further requires M = T Then, following the logic above, we can derive the following terms for the Jacobian:
(F,(W) = (ōje Eis + 85h Eie + diz Ejt + de Eej) Mij, aW
(30)
(Fr (ōze Eik 55k Exe + Sie Eyt + der Eij) Mki-
(31)
aW.
Then, the symmetry condition yields two constraints:
i = and j + f, then Eej Mdj- = MM Σej. so for each row i and any column pair (j, E)
with Σe, * 0, one must have Mij Mie
If j = and i * k then E. M,j .Mj Eut, so for each column j and any row pair (i,k) k)) with Eis * 0, one must have M.j = MMj-
In the generic case Egp of for all i,j, these conditions force M be either the all-ones matrix 11 T or its negative. Hence, in the case of symmetric matrices, gradient flow can be realized only for purely Hebbian or purely anti-Hebbian networks. Indeed, if M 11 the learning rule in Equation (24) is symmetric, so that any symmetric initial (0)) remains symmetric, and the learning rule can be written the following gradient descent:
W ~ E EW + WE - Vww(-Tr(NN WEW
(32)
If M = -11 the minus-sign simply flips the gradient.
26

---

A.3 Hebbian plasticity in excitatory-inhibitory networks
We will next follow a similar logic to demonstrate that the learning rule (Equation 2 in the main text) cannot be written as gradient descent in general:
Tw W Tf + WD + ff DWT,
(33)
+1,
if neuron is excitatory,
where D RRXNN is diagonal with Da 1-1, neuron is inhibitory.
Proof. As before, define E = T (or ( for varying inputs). The first term written as a negative gradient:
33 can be (-Tr-r WT)). Since Equation 33 is symmetric,
we may restrict attention to W = W otherwise, requiring symmetry of the Jacobian alone would force Σ 0, similar as before. To test for a gradient flow, we again inspect the Jacobian symmetry condition, where we now consider the following function corresponding to the second and third terms of the learning dynamics:
F(W) = W EDWT.
(34)
direct calculation gives:
0Fij 8sk de Dej + die dx Eas + 83z de Eie + 85e du Exee
(35)
OF Sut dj Eiee + 6e d Exe dn d3 Ens + des de Ess-
(36)
aW,
In particular, setting i k and * l results in the following constraint:
de Eej d3 Eej
Vj,
(37)
Thus, whenever Σg, * O, one must have de = d,. the generic case E,j A for every pair (). this forces D = +IN and so eliminates excitatory/nnhiitoryy diversity. Therefore,in general the Hebbian learning rule excitatory-inbibiorry networks cannot correspond to gradient flow. O
Remark. Note however, that if inhibitory neurons receive zero external input (so that corresponding rows and columns of Σ vanish), the constraint in Equation (A.3) is satisfied without collapsing D to +IN, and the curl terms are nullified. In Section A.4 we will demonstrate that previous work proposing seciaator-inhiiiton
circuits that can optimize a similarity matching function (Pehlevan et al. 2015) fall into this category, where the curl terms are climinated by the choice of structure of the network.
A.4 Obtaining gradient flow in El networks by nullifying curl terms
As in [Pehlevan al. 2015), we consider neural network made of an excitatory feedforward layer connecting the ipput to hidden layer of excitatory neurons, which themselves form a recurrent loop with inhibitory interneurons. This architecture commonly used for feature extraction biologically
plausible neural networks [Foldiak 1990 Rubner and Tavan 1989 Kung et al.) 1994 Leen 1990).
Pehlevan et al. 2015] showed that Hebbian/anti-Hehie
plasticity in such networks can optimize similarity matching function. Here we take the inverse approach: instead of asking what is the neural architecture and learning rule that can support the optimization of given cost function, impose neural architecture equipped with a Hebbian learning rule and ask what objective is optimizing.
Grouping neurons by their structural role, the activity of the full network can be written y [YFF, YReeE, YReel T. Using the framework in Section 2 of the main text, the neural dynamics are given by
r -y + WDy +f,
(38)
where the weight matrix (following the circuit structure in Pehlevan et al. (015 can written in block form as:
W
RoCE 0
0
WoeeI-RocE
(39)
WReCE Reel
27

---

where the matrices correspond to non-existinn non-existing synapses. The correlation matrix of the input is
ff T
EF
Σ =
0
0
0
0
(40)
0
0
only the excitatory neurons receive external input, and the D matrix is
0 1 0
D
0
(41)
0
Taking the Taylor expansion of the neural dynamics at steady state around W gives y* = (I
DD (I WD + (WD)² + (W (WD)&& O(W"))f. Under Hebbian plasticity, and noticing
that DE = ED = Σ due to the lack of input to inhibitory neurons, we obtain the following weight dynamics:
TW W yy R E + WS+ WW + (WD) + WEWT =(DWT)
(42)
WWE(DWT)& "
(43)
we take close look at the structure of each of these terms one-by-one, in comparison the block structure of W in Equation (A.4). we can observe that many of the terms are effectively nullified as they predict weight updates for synapses that structurally do not exist.
Σ maps onto non-existant FF-+FF synapses and is therefore nullified.
0
. -WE =
W WFF-
0
0
maps onto FF-*ReeE synapses and is maintained.
0
ET WT
0
FF-+RoB
EWT
0
0
maps onto nonexistant RecE->FF synapses is nullified.
0
(WD))
0 0
maps onto nonexistant FF-+Rec
WReCE Reat Fr Reck PF
0
0
synapses and is nullified. Its transpose EDW
T2, also nullified.
WEWT 0 W Reaf F FooR
0
0
= Enee is the covariance matrix of the
0
recurrent excitatory neuron activities driven by the feedforward input. too maps onto non-existant RecE - RecE synapses and is nullified.
0 FF WFF RecF WRocF Reel
E(DWT>2
0
maps onto non-existant RecI->FF
0
0
0
synapses and is nullified.
0 ())
(WD)3 - CecReeeeeooo - WFF FROE CFF
0 is maintained.
0 0)
(WD)SEWT WDEREC
O
0
0 WReeh RRec WFF
W.T FFO-
0
is maintained.
0
0
WE(DWTJR ERSE DWT
W WFY-RROLt Dpp WT--WT-- WT RoceE-Roel
0
is maintained.
28

---

- -FF FF- Rock W
REcE Roed
W
W
E(DWT)$-
0
RocE
0
maps non-existant
0
RecE-+FF synapses and is nullified.
Note that terms of order 0 and 2 nullified and we are left only with terms of order 1 and in W. Collecting the non-nullified terms results in the following weight dynamics:
W = WE + (WD) (WD)>EWT WE(DWT)
(44)
Since the update for Weed--Reet is the transpose of that of WReeE-Reed one can assume that these two matrices will converge be the transpose of one another, and after collecting terms the update rule can be expressed as:
Tw = V Tr (MMee)eeee).
(45)
Here, M W Rect W RecE- excitatory neurons, and Erec the covariance matrix of the recurrent excitatory neuron activities driven by the feedforward input. Minimizing this energy function achieves two principal goals:
denotes the disynaptic inhibitory feedback from the recurrent
1. First, the term 'r (Eise) promotes the maximization of the total variance captured by the excitatory neurons, thereby driving the feedforward weights to perform a PCA-like extraction of high- variance features from the imput covariance Eyp.
2. Second, the term Tr (MEEe)) is minimized. The inhibitory feedback matrix M learns the covariance structure of Erec- This second term will therefore reduce the learning of features already learned.
Overall, this energy function drives the excitatory population to acquire a progressively decorrelated set of features.
This architecture, equipped with Hebbian learning rule, allows its learning dynamics to be expressed gradient flow. This result is consistent with previous work in which this architecture was previously derived from normative principles [Pehlevan et al. 2015 Kung et al. 1994). although the learning rule and objective here slightly different. However, introducing input-to-inhiibiorry synapses as well as recurrent excitatory-to-excitato or thubitory-to-inbiiitory connections would introduce curl terms in the learning dynamics, which could no longer be written as the gradient of any function.
29

---

B Large networks
In this section, we provide detailed analytical derivations regarding a two-layer linear neural network with M input neurons, N hidden neurons, and scalar output, and whose weights evolve under curl descent (Equation 5 in the main text). In section B. we provide an expression of the Jacobian of the weights' dynamics of our system in the general case, and then use this expression to derive its eigenvalues on two types of critical points: the origin saddle (section ( and the solution manifold of gradient descent (section B.3) Finally, in the latter case, we leverage random matrix theory (section B.4) to characterize, in the large M, N limit, when the support of the Jacobian's s eigenvalues on the solution manifold crosses the origin, characterizing the phase transition from stability (exclusively negative eigenvalues) to instability (including positive eigenvalues).
B.1 Full derivation of the Jacobian
Consider Wi RR- matrix and W2 R matrix, D, and D2 diagonal matrices of respective sizes M x M and N x N, with diagonal elements d1,; = 11 and d2, = #1.
As in the main text, the curl descent learning dynamics are given by:
Wi W (s - W2W1)D1
(46)
W3. (s W221)W,, D2
(47)
We will use == 8 W2W, E RM, ve E RNM and ve2 E B., The full Jacobian expression can be broken down to block matrix with four blocks:
Full Jacobian expression.
Jn1
J12
J =
(48)
J21
J22
We proceed to compute the expression of each of these blocks.
Jn computation of aw: aW1*
W1.5 E2d5E,d1
(49)
W1.0 W2.1(53
Σ W2,1 W1,j)a)
(50)
WW...
W2,W22p (d1
(51)
aW 1
ow,
J11 ow,
(WT W3) & D1
(52)
J12 computation of aw: ZW3
aW.
W2, Wi.nddd + Shi 81 (8-->D EWW.
(53)
aWen
-niEEdis W2, Winsdii
(54)
aw, aW,
SniEEdis 22, W1,nydi,j
(55)
(1))
30

---

J21 computation of aw3 OW;
M
W2,1 = Σ R=1
E& 1. with Ex = 86 Σ W2,tW
(56)
OW2e
8ep Egg W2.eWie
(57)
@W1,pq
OWW2
(J21le((p4)
= =depEgd2 W2,e
(58)
@WW1
2.(0g)
Jze computation of aW3 aW2
M Eu R.Winnke aad
W2.
with Eg 8k Σ W2,tW2,th 2,th
(59)
oW2t
J22 = 222m
Winsa
W Jen
(60)
k
(61)
The cigenvalues of this Jacobian are given by det(J - XI) O.
(J11 (JI)) (J11-AlINa
J12 Jn Ala
( D
A
Al
(62)
J21
(63)
B.2 Evaluating the Jacobian eigenvalues at the origin
At the critical point (W1, W22 (0, 0), we have W, 0 ED, and W₂ EO D2. with E
s W2W1 = s. The Jacobian blocks become
Jn ONMXNM
(64)
[J (1) = dnis,d1.j
(655
[Jzn (ull..pp)) 81pSyd2.1
(66)
Jz2 ONNN N
(67)
And therefore
N Jz1
J12
J(0,0) - AI AI=
(68)
-AIN
If M 1, then NM A N and det (J 0) 0. meaning that À 0 is eigenvalue of the Jacobian.
For the nonzero eigenvalues, the Schur complement yields:
det(J - AI) = det(-AINNA) det (-XINN --I---- J12)
(69)
= =M det (-ATNN + x Jzn J21 J12
(70)
= -(-A)NM-- det (A2IN - J11
(71)
31

---

(J21J12hu =
Σ
(72)
1/
d2,101h
Σ
d1.83
(73)
M
Σ
D2
(74)
th
Hence have
M
det(J(0,0) XI) (-A)MNN-N
II
X2
d2
2
d1
(75)
j=1
Therefore the origin, which is the only saddle point in parameter space, has MN - zero eigenvalues and 2 non-zero eigenvalues:
Σ M
j=1 d
if d2,
A
(76)
j=1 d1
if d2,i = -1
The origin is turned into saddle-center.
B.3 Evaluating the Jacobian eigenvalues on the solution manifold (E = 0)
The Schur complement formula gives det(J/ AI) det (AA)
CA- provided that is
invertible. We will investigate the stability of the fixed points of the dynamics verifying E = 0.
We compute the determinant of matrix of each block A; - W2) applying the matrix determinant lemma results
-WW, W2) D1 XTMN by computing the determinant Allx for [1; MJ. Noticing that W, W2 is rank 1 and
det(A,) = (-1) Wow: d1.1)x~1
(77)
= (-11) (1+ (A+
(78)
Therefore
M
dct(A) = - 1) N N- (A d2..l|W211?)
(79)
M
= MN M M(N-1)
II
(A + d1, 12)
(80)
1=1
We now compute the Schur complement D - CA-1B, starting with the A-1 matrix. Using the Cherman-Morrison formula:
di d..W, W₂
IN
A
(81)
X(A + d2 (1
X
Importantly, W an eigenvector of the A, matrices:
W
4 W.
(82)
A dr,a W2
32

---

Therefore simplifying the calculation of A B, for one block j:
We.WWiM
[Aj'B]] A + d1 W2ll2 dij
By]
(83)
Yielding
W2,iWi,
B]
di.
(84)
(i))
X d1 [2
Multiplying on the left by [C],(pg)
222,p Wi, Widd yields:
M
W2, W2,:W1.N Winj
[CA-
Bl
Σ
W2 1.,d2.1
d1,
(85)
X + d1,j [W2]
d2.j
=|
Σ
W + d1 W2|l
WTT 1igh
(86)
j
(87)
Finally,
M
d1
- B) =[-D2WiW, - ALNluR un + +1122|12d2, Σ
W Wi, A + di
W
(88)
W2
M
d1 W2
A x d 1211
= =d + d2,1
W1.15
W
(89)
A di W 12
A + di,3 |/|
1,5h
j=1
=dddn Ad2,1
W1.15 X + d1, W2
W..ji 1.jh
(90)
->(DDWWWWT + +l)
with A = diag
(91)
A d W2l2
(92)
And we have, as for the expression of the determinant of CA-1B
det(D - CA- det (-A + Iw))
(93)
(->)-" det (IN D2W7AW,7)
(94)
(-A) det (IM + WTT D2W1A) as
det YX)
(95)
= =--A)" det ((A-++ + W,T D2W1) A)
(96)
--AA det (A-1 + WWT D2W1) det (A)
(97)
Noticing that (A) (-I)MN 11 ( det(A) -1, the full determinant of (J - AIl) now reads:
det (J AII (-1) MOOA:NNAMMNN---. MNN+N-M det (diag(A + di, + W, D2W;
(98)
The eigenvalues of the Jacobian solution points are determined by the equation del (J - All) O. One can therefore look at the conditions on the network's architecture parameters M and N such that the solution manifold remains stable upon the introduction of rule-flipped neurons in either the hidden layer the readout.
33

---

B.4 Evaluating the stability of solution points (Jacobian eigenvalue distribution for E 0)
In the following, we determine the conditions on the architecture parameters M, N and the plasticity parameters conveyed by D1, D2 needed to ensure the stability of the solution manifold determined by E We will separate two case scenarios: one where Dj has mix of +1 with D2 TN. and inversely where Dj = M and D2 has a mix of +1.
B.4.1 D1 with mixed signature and D2 IM
In that case, we have from equation 98
(-1))==-1) del (J- AT) = (-1)NMr+N A MMN+N-M del (diag(> + d1,5 (W2)l3) W, Wi
(99)
The Jacobian therefore has MN + N M null eigenvalues and the others verify the equation
det diag(A + d1,; [W2l + W, Wi
(100)
det (-XIIM (diag(d1,//lW2/PP>> w, Wi)
(101)
That is, we would like to determine the eigenvalue distribution of the above defined X RRM matrix which is a sum of a diagonal indefinite matrix with a Wishart matrix.
To proceed, will assume that:
W1.99
iid. N(O. 1/M),
idd W2,1
NOO, 1/N)
(102)
The 1/M and 1/N scaling for the W1 and W2 matrices account for He initialization He et 2005 and ensures that W1 Wi has non-divergent spectrum without any further rescaling.
For large N, the law of large numbers gives ||2 ss 1. Without loss of generality, let
m_ M
D1 diag +1
=
A:= (1-ax)) = 1122a E [1,1,
(103)
Define the ratio
M
(104)
The object of interest is X == Dí W, R A in the joint limit M,N with fixed
Cauchy transform of Di. For 2 $ {+1},
Gz) = 1 1 0A
Oh.
=
(105)
2+1
22
Blue transform of D,. Set w GD, (z) and invert:
w 22 1) = 2 + A w2 2 (w + A) = 0.
(106)
Solving this quadratic equation for z and choosing the branch with BD, (w) ~ 1/w as w + 0 yields
+ 4w A)
Bp, (w)
(107)
2w
34

---

A(w, 2x)² R(w). R-transform of D1. The R-transform is defined as R(w) = B(w) 1 Hence
V 1 + 4w w A)
RD, (w)
(108)
2w
Marcenko-Pastue law with variance 1/M. Let S = W WT" Note that here the entries have variance 1/M, and not the usual sample-covariance prefactor 1/N. limiting law
Marčenko and Pastur 1967 give
Ps MP(c)..
support [(1 -1--v) Vo/ (1+ (L+ /c]
(109)
R-transform of S W, WTT. Using the property of the R-transform Rax(w) aRx (Rx((a)) with we obtain
1
Rs(w) c(1 w)
(110)
Asymptotic freeness of Dj and S. = WWT is orthogonally invariant, i.e. US & for any deterministic U E (M). because W Gaussian. An orthogonally invariant random matrix is asymptotically free from any deterministic matrix Collins and Male 2014]. Therefore
D and S
are free
(111)
Combined R-transform of X. From I08
110
Rx Rx(w)= = RRD + Rs(w)= /1 + 4w(w A)
(112)
2w
c(1 w)
Blue transform of X.
Bx(w) = + Rx(w)= 1 v 4wo( + A
(113)
w
2w
c(1 w)
Implicit equation for the Cauchy transform. definition of the Blue transform:
Let Gx(z) be the Cauchy transform of X. By
Bx(Gx(2))= =2
(114)
Write w == Gx (2) and 2 = x E R (real spectral parameter). Introduce
2(w,z) 2w
c(1 dl-ww) ]
R(w) 1 4w 4w
(115)
Equation 113 is equivalent to
(116)
Multiply 116 by (1 2 to clear the denominator. Collecting terms produces the quartic polynomial
(ww) == a + azwu + azwa + aiw + ag = O,
(117)
with coefficients:
04 = 4 (x² - -1),
(118)
03 = -4Ac Scx² - 4cr + 8c + 8x
(119)
02 S + 202 + 8...x - 462 Scr 4e +
with A 1 2ahh
(120)
4Ac --eer +4
61
(121)
00 = 0.
(122)
35

---

Differentiating this polynomial gives
(w) 11( 1)w² -12Ac - 24cT² 12cx + 24c + 24x
w²
(123)
16Ac xc²x² + 16c - 862 - 16cx 8
-4Ac 4er +4
(124)
with A 200
Support of the spectrum 117 i.e.
An endpoint of the support occurs when w becomes a double root of (w))) 0.
(w))= 0,
(125)
Eliminating w from (25 yields quartic polynomial in x; its real roots appear pairwise. The eigenvalues support will therefore be the union of at most two intervals.
When the support yields exclusively negative eigenvalues, then the solution manifold is stable. The theoretical boundary for the solution manifold stability corresponds to when the support crosses 0, that is when the Jacobian on the solution manifold starts having positive eigenvalues.
B.4.2 D2 with mixed signature and D1 TM
The stability boundaries for the complementary case, in which we instead vary (keeping on = 0), can be found using the method of [Kumar and Sai Charan 2020).
36

---

C Simulations for tanh networks
In the main text, showed that in linear networks, the learning dynamics beyond the stability boundary yielded transition to chaos when rule-flipped neurons were introduced in the hidden layer, destroying performance. We also showed that when destabilizing the solution manifold by introducing rule-flipped neurons in the readout layer, the network still managed to reach small testing error. this section, we provide additional simulation results for nonlinear tanh networks. The qualitative behavior of the tanh networks is similar to that of the linear ones: we recover the transition chaos (Figure ( and the ascend then re-descend mechanism (Figure 9 and the phase diagrams
show similar trends although the boundary are shifted to lower values (figures 6 and g
Fraction of rule- fee-fiipped neurons dn
Test-error heat-map
b)
Fraction rule fllppedppdddeee neurons dn
Test-error heat-map
10-1
0.8
0.8
10
0.6
11022
test error
0.6
test error
0.4
0.4
10
10-3
0.2
9--1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
102
ratio c= M/N (inputs/ hidden neurons)
ratio c= M/N (inputs hidden neurons)
Figure 6: Hidden layer phase transition for tanh networks. a) Test error as a function of the compression ratio and the fraction rule-flipped neurons oд (averaged over 10 seeds). Black curve: analytical stability boundary derived for linear networks. b) Close-up for e E [0.1, 0.7]. Compute resources: hours 500 CPUs (local cluster).
a)
b)
C)
M
Autocorrelation
Weights
0.0
PC2
-0.2
-2
-0.4
-0.50
2000
4000
6000 Epoch
8000 10000
20000 40000 60i0 80000 100000 Lag (epochs)
0.0
PC1
Figure 7: Hidden layer curl terms lead chaos in tanh networks. Example simulation tanh network with Nov 110 neurons, 0.8 and ay = 0.6. a) Weight dynamics function of the epochs. b) Weight autocorrelation functions. e) Weight dynamics projected on its first two principal components.
37

---

Test-error heat-map
b)
Max error heat-map
Fraction of rule-fiiped neurons dr
100
Fraction of rale- rule-fllemmn flepeldpdd neurons dr
100
0.8
10
0.8
10-1
0.6
Lest urroN
10-2
max DETOR
10
0.2
90-2
10-1
10
10
90-2
10-1
1
10
10
ratio c= M/N (inputs / hidden neurons)
ratio c= M/N (inputs / hidden neurons)
Figure 8: Readout layer phase transition for tanh networks. a) Test error as a function of the compression ratio с and the fraction of rule-flipped neurons or (averaged over 10 seeds). Black curve: analytical stability boundary derived for linear networks. b) Peak learning error (maximum over random seeds, initialized near the solution manifold). The black curve shows the analytical boundary derived in the linear case. Compute resources: 6 hours on 500 CPUs (local cluster).
b)
10-4
10-1
Train error
10-9
Test error
10-1
2
10-14
10-12
PC2
10-18
10-24
10-28
10-21
10a
109
10
10>
10
-4
-2
timestep
Epoch
PC1
d)
1.00
Autocorrelation
0.75
Weights
0.50
0.0
0.25
-0.2
0.00
-0.4
20000 25600
5000 10000 2000000005000 Lag (epochs)
Epoch
Figure 9: Readout layer curl terms in tanh networks result in low error even when the solution manifold is unstable. Example simulation of tanh network, with Not = 110 neurons, c = 1 and ar 0.6. a) Training error for Curl descent initialized a small distance away from the solution manifold by adding 10-15 perturbation on the weights (light purple), and training error for gradient descent, initialized randomly (dark purple). b) Same for testing error. e) Weight dynamics projected on its first two principal components. d) Weight dynamics as a function of the epochs. e) Weight autocorrelation functions.
38

---

D Faster convergence for ReLU networks
To verify that the accelerated learning observed with curl descent tanh networks is not restricted to sigmoidal activation functions, we replicated the experiments obtained for tanh networks in feed- forward architectures whose units employed rectified-linear (ReLU) activations. We used the same student-teache set-up (M = 100 inputs, N 10 hidden units), and identical parameters. The faster convergence effect on ReLU networks was smaller, hence the 40 random seeds for statistical significance. The results are shown in figure T0
b)
to 10 testing error
10"
Gradicat Meditted Gendion Descent
Number of epochs
Number of epochs
Test errr
to 10-2 11011ttt tasting
100
10-2
10'
10'
10
Epoch
500
1000
1500
2000
10
Training set size
Weight: initialization scale
Figure 10: Relu networks: curl descent leads to faster convergence in broad parameter regime.
a) Test error for curl descent and gradient descent (NV 1400, weight initialization scale 2;
error bars indicate + sem, averaged over 40 random seeds). b) Convergence speed of curl descent
and gradient descent as function of training set size (weight initialization scale 2, P < 0.05 for Nirain 2 1000). c) Same b) as function of the weight initialization range (Nurain 10000, p < 0.05 for weight initialization scales 3, 4, 6, 7 and 8). Compute resources: 24 hours on 500 CPUs (local cluster).
E Faster convergence without the weight renormalization step
a)
b)
to 10 testing error
10
to 10-1 testing error
Number epochs
Number epochs
201
30'
10a
10
1000
1500
2000
Student weights
Training set size
initialization scale
to 10-3 testing error
Number epochs
Grutscar deccet
Log(test error)
-4
velghr
-6
101
-0
vit moggt
6.0
0.5
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Figure 11: Nonlinear networks: curl descent without renormalizing the weight matrices at each time step leads to faster convergence broad parameter regime. a) Convergence speed curl descent and gradient descent function of training set size (weight initialization scale 2). b) Same as e as a function of the weight initialization range (N = 10000). e) Log of the test error after 20, 000 epochs as function of the teacher weights initialization scale. d) Convergence speed as a function of the fraction of rule-flipped readout neurons. Compute resources: 12 hours on 500 CPUs (local cluster).
39


Reviews:
[{"review_id": "XskOP2OVKe", "reviewer": "Reviewer_DQUi", "summary": "The paper introduces Curl Descent, a theoretical framework for analyzing and leveraging non-gradient learning dynamics induced by biologically plausible, sign diverse plasticity rules. By decomposing learning updates via the Helmholtz decomposition into gradient-conforming and curl components, the authors formalize how heterogeneous synaptic plasticity; specifically the coexistence of Hebbian and anti-Hebbian rules: generates structured, controllable non-gradient dynamics. Using random matrix theory and explicit Jacobian spectral analysis, the paper quantitatively characterizes how the magnitude of these curl terms governs both the stability of learning trajectories and the geometry of the solution manifold. The results reveal that expansive architectures, where hidden layers have greater dimensionality than inputs, can inherently buffer against curl-induced instability, while networks operating near critical curl magnitudes exhibit rich, nonlinear learning dynamics.\nThe paper demonstrates that curl components are not universally detrimental to learning. Through an analytically tractable two-neuron system and larger simulations, the authors show that moderate curl magnitudes can facilitate beneficial dynamics, such as detours through unstable regions of parameter space that ultimately accelerate convergence or improve escape from saddle points. This phenomenon is particularly evident in non-linear networks, where introducing controlled curl terms reduces the number of training epochs required to reach low error levels, outperforming strict gradient descent under certain conditions.", "questions": "a. How does the interplay between the network’s expansion ratio (c = M/N) and the magnitude of the curl component quantitatively affect the phase boundaries for stable versus unstable learning dynamics in the random matrix regime, and can this relationship be generalized beyond the specific student-teacher model analyzed?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\nThis work presents a well constructed and theoreticaly rigorous framework for understanding how biologically motivated, sign diverse plasticity rules induce curl components within learning dynamics. Through a principled application of Helmholtz decomposition, the authors distinguish between gradient conforming and non-gradient components of weight updates, providing a clear mathematical basis for analyzing non-gradient learning dynamics. The use of random matrix theory to characterize the stability properties of these dynamics, particularly through explicit derivations of Jacobian spectra and phase boundaries, reflects a commendable level of analytical depth. These results give concrete, testable predictions about how curl magnitude, network architecture, and plasticity sign diversity interact to govern learning behavior.\nBy linking curl dynamics to well-established biological principle; such as Dale’s law and empirical findings on heterogeneous long-term potentiation and depression—the authors successfully ground their mathematical results in experimentally observable phenomena. The inclusion of toy models, such as the analytically tractable two-neuron system, effectively conveys key intuitions regarding how curl components can introduce detours in parameter space that initially destabilize but ultimately accelerate convergence. These insights challenge prevailing assumptions in machine learning that gradient-following updates are strictly necessary for efficient learning and open promising avenues for exploring biologically inspired, non-gradient learning rules in artificial systems.\n\nWeakness:\nWhile the theoretical framework is compelling, certain technical claims, notably those regarding the uniqueness of the gradient term within the Helmholtz decomposition and the determinant-based stability conditions for larger networks, are only partially elaborated or deferred to supplementary material. Although the supplemental derivations appear sound upon spot-checking, presenting these results more completely within the main text, or providing an accessible computational (jupyter) notebook, would greatly improve transparency and facilitate verification by others.\nWhile the simulation results qualitatively support the theoretical predictions, the empirical scope remains narrow, focusing primarily on small-scale synthetic tasks without extending to real-world benchmarks or diverse architectural settings. The absence of classification tasks, recurrent models, or larger datasets makes it difficult to assess how robust or generalizable the observed benefits of curl dynamics truly are.Lack of an accompanying, open source code release and incomplete reporting of experimental details;such as the implementatino details. Addressing these aspects would significantly improve the paper’s reproducibility and ensure that its insights are more easily transferable to practical machine learning and neuroscience research contexts.", "was_revised": true, "final_justification": "The authors have clarified my questions and concerns, and hence it is a favorable score.", "rebuttal": "We thank the reviewer for their constructive comments. We respond point-by-point below.\n\n- ## Uniqueness of the gradient term:\n\nWe emphasize that the Helmholtz decomposition is not used explicitly in our work. We mention the decomposition as an analogy for the existence of both gradient and non-gradient terms in the dynamics. We prove that curl descent dynamics is a non-gradient system without explicitly deriving a decomposition into a gradient field and rotational field. As such, we make no claims on the uniqueness of any gradient or non-gradient terms, and only require that the non-gradient term is non-zero. \n\n- ## Stability conditions for large $N$:\n\nFor the stability conditions for large networks, we first derive the characteristic polynomial of the Jacobian of the curl descent learning dynamics, which we evaluate along the solution manifold. The eigenvalues of the Jacobian matrix of a dynamical system evaluated at a certain point are informative about the local stability at this point: if all eigenvalues’ real parts are negative, then the point is stable, otherwise it is unstable. We therefore use random matrix theory to determine when the spectral support includes positive values (in the limit that $N \\to \\infty$ and assuming i.i.d. Gaussian $W_1$, $W_2$). We do this by considering two cases: sign diversity in the hidden layer (i.e. in $D_1$) or in the readout layer (i.e. in $D_2$). In the readout layer, we use tools from free probability theory to derive the support boundaries of the spectral distribution as the roots of a polynomial that can be evaluated numerically. In the hidden layer, we can directly apply the method of Kumar and Sai Charan, 2020 which directly provides the expression of the polynomial whose roots determine the spectral support’s boundaries. All derivations are in the appendix due to limited space in the main text. However, we sympathize with the reviewer’s concern that the basis for these derivations may be obscure. We will therefore include a brief sketch of the derivation in Section 4.2, and we will include additional details in the Appendices to walk the reader carefully through the derivations. Additionally, we will provide a jupyter notebook comparing the empirically obtained spectra of the random matrices to the theoretical predictions.\n\n- ## Generalization of curl descent benefits:\n\nWe fully agree that the empirical evidence presented in this work does not demonstrate the generalization of curl descent gains to more complex architectures or tasks. The primary contribution of this paper is to give a first treatment of non-gradient dynamics, in an analytically tractable setting.  Our results therefore show that within this setting, curl descent can enhance learning within a limited region of structural parameter space that can be analytically identified. To date, we lack analytical results that would guide the correct parameterization of more complex networks on real-world datasets. \n\nHowever, as suggested by reviewers mR3k and 2UnU, we will improve Figure 5 through additional empirical results regarding the effect of curl descent dynamics under other conditions, while still operating within the student-teacher setting described in our work. We conducted further numerical analyses on two recurring concerns from the reviewers’ feedback: the generalization of curl descent benefits to other architectural parameters and task complexity.\n\n### 1. Architectural parameters \n\nWe conducted new simulations to investigate the effect of increasing the fraction of rule-flipped rule neurons in the readout layer for three compression ratios (0.1, 1 and 10). We found that the benefits of curl descent, as well as the range of rule-flipped neurons fraction improving learning, were more pronounced in networks with high compression. \n\nRegarding the introduction of rule-flipped rule neurons in the hidden layer, we did not find any structural parameters that lead to improved performance compared to gradient descent, consistent with our analytical results in linear networks (Figure 3a)\n\n### 2. Modulating task complexity\n\nWe have added new simulations where we parametrically increase the complexity of the task by expanding the teacher’s initialization range. This is known to increase the complexity of the teacher representation and, consequently, the difficulty of the task the student must learn (Poole et al., 2016). These simulations reveal a threshold initialization range above which there is a sharp drop in convergence speed for both curl descent and gradient descent, as the dynamics in either case are stuck in a saddle. Above this threshold, both rules converge at the same (slow) speed. Below this threshold, curl descent converges faster than gradient descent. \n\n- ## Transparency and open source code:\n\nWe wholeheartedly agree with the reviewer’s concerns regarding transparency and reproducibility. We will make all code used for the simulations publicly available on GitHub, including in the form of jupyter notebooks. \n\n- ## Generalizations of the phase boundaries:\n\nWe used the approach described above to solve for the phase boundary between stable vs. unstable learning dynamics, as a function of $c$ and the number of sign flips (which is a proxy for the curl component).  We focused on two-layer feedforward student-teacher networks as they are a widely studied framework for mathematically tractable analyses of gradient flow. This allowed us to derive precise theoretical insights as a first step towards understanding non-gradient learning dynamics. Our analyses could in principle be extended to other architectures, including deep networks or with convolutional layers, although the derivation of the spectral support will surely be more onerous. Extensions to recurrent architectures are more challenging, as the interplay between recurrent dynamics and learning dynamics could generate new phenomena that are beyond the scope of this study. Nevertheless we acknowledge that our study is just a first step to understanding non-gradient terms. We will highlight these limitations in the discussion as promising directions for future work."}, {"review_id": "vL6yoiW5rd", "reviewer": "Reviewer_2UnU", "summary": "This paper is a normative study on synaptic plasticity in neural networks. Normative approaches have mostly focused on learning rules that minimize a given cost function (i.e. that follow a gradient). Conversely, the plasticity rules that are not derived from a cost function are typically considered from a purely empirical or numerical point of view. Here, the authors provide an analytical perspective on the latter group of rules. They first prove that “sign diversity” (in the plasticity rule: co-active Hebbian + anti-Hebbian plasticity; or in the neuron types: excitatory and inhibitory neurons) generate learning dynamics that do not follow any gradients.\nThe trick to nevertheless provide a normative treatment of such systems is to use a teacher-student framework in linear feedforward networks with a proportion of synapses doing gradient ascent while others do gradient descent. The learning dynamics can then be broken down into a classic gradient-descent component and additional “curl” terms that affect the learning dynamics depending on parameters such as the network architecture and the proportion of flipped synapses. The authors show that curl terms can change which solution from the gradient-descent dynamics is reached, sometimes speeding up learning.", "questions": "1/ Each plasticity rule/cell type could be understood to optimize its own gradient/objective. Even though the whole system doesn’t optimize a single objective, could it still be understood as some competition across neurons for different cost functions? This would somewhat nuance your claim challenging the prevailing view of gradients for synaptic plasticity.\n\n2/ Over-focus on one specific instantiation of Hebbian and anti-Hebbian learning In section 2. What about other rules? For instance variations of hebbian/anti hebbian. I don’t think you need to add anything in main, but maybe point to a supplementary paragraph that suggests that the findings are not overly rule-specific. Right now, this makes your results appear to have a narrower scope than they surely have.\n\n3/ Your motivation for adding different plasticity rules in the same network are cell types, which are typically defined by their connectivity patterns to other cell types and their unique cellular properties. Your model is thus making a rather strong assumption of only changing the rule. I understand an analytical treatment of such networks with more complex initial connectivity is out of scope, but maybe you could provide some numerics? Another way of asking the same question is: What if the identity of the flipped synapses was not random: can you show further/more robust benefits of curl terms in such networks? You could take inspiration from/recover classical synapse types connectivities? (local PV/SOM interneurons for instance)\n\n4/ To make this work more impactful and strengthen the claim that curl terms are more than an “approximation error” to gradient descent and deserve proposer attention in the future, I think that figure 5 needs to be expanded. I understand this is out of scope/hard to do analytically, but some additional simulations would be welcome. E.g. recurrent networks/a more “real-world” task than the teacher student framework.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Overall, I found this paper clear, well-explained and nicely presented. The model chosen is simple, at the limit of being simplistic, but just complex enough to obtain interesting conclusions about the problem at hand. As such I find it a very elegant and important work in the current scientific scene around synaptic plasticity, although it could be attempting to formulate more precise predictions towards the end.", "was_revised": true, "final_justification": "Overall, the authors have addressed my remaining concerns (clarity, discussion and additional experiments). I believe this is a high-quality contribution to the field and recommend acceptance.", "rebuttal": "We thank the reviewer for their constructive comments. We respond point-by-point below.\n\n## 1. Competition across neuron-specific cost functions \n\nThis is a great point and the intuition is absolutely correct: each rule-flipped neuron is individually trying to climb the gradient, so in a sense they do have their own cost function, which is minus the mean squared error. These neurons can therefore be interpreted as not just competitive, but adversarial. With that perspective, we find it all the more intriguing that not only are the learning dynamics robust to such adversarial neurons, but that these adversarial neurons can even speed learning in certain situations. \n\nThat adversariality makes curl descent different in spirit from a network in which two pools of neurons have competing cost functions that are separately normative (e.g., error minimization and homeostatic regulation). In other words, it isn’t a case of finding a compromise between slightly misaligned objectives: in curl descent, the neuron-specific objectives are fundamentally incompatible.\n\nThus, while curl descent requires  computing gradients, it is not implementing gradient descent, not even for the individual populations. By having two populations of neurons with opposite cost functions,  the two cost functions can’t both decrease. \n\nWe will add this important and nuanced point to the discussion, including the adversarial interpretation of curl descent dynamics, and the clarification that curl descent challenges *gradient descent* as a unified framework for learning, but still relies on *gradient computation*.\n\n\n## 2. Other learning rules\n\nWe agree that this section was overly focused on a specific form of Hebbian/anti-Hebbian plasticity. To give the simplest example of where non-gradient terms may arise, we used the product of pre- and post-synaptic activities which is common across variants of Hebbian plasticity. As suggested by the reviewer, it is straightforward to extend our analysis to these variants. One example, pointed out by Reviewer mR3k below, is adding additional terms to avoid weights decaying to zero or exploding to extreme values (e.g., in Oja’s rule, BCM, covariance or homeostatic rules). In our response to their comment, we provide an example in which including a leak term to stabilize the weights can be seen as a regularization term added to an objective function, and therefore does not affect the presence of curl terms. The same reasoning applies to other variants of Hebbian plasticity, which are often explicitly derived from the gradient of an objective. \n\nWe acknowledge that section 2 in its current form, and the subsequent use of the term ‘Anti-Hebbian-like’ neurons in following sections, can be overly simplifying. We will address this by moving the ‘Hebbian/anti-Hebbian networks’ paragraph to supplementary materials (only focusing on the EI network example in Section 2), with an additional paragraph to explain how it can be extended to other variants of Hebbian plasticity. We will also change terminology from “anti-Hebbian-like” neurons to “rule-flipped  neurons”.\n\n## 3. Cell types with non-random patterns\n\nThe reviewer is right that assigning random rule flipped rule neurons is an important assumption in the paper. If the cell types are defined so that their plasticity rules *and* their structural connectivity is different, this can significantly affect the results. In particular, we showed an example in Appendix A.4 that correlating the plasticity rule to connectivity motif can result in gradient flow, if the structure is carefully chosen so that the curl terms are effectively nullified. This is the case in a previous proposed normative model of EI networks learning PCA (Pehlevan et al., 2015), which was the basis for that analysis. \n\nThe reviewer instead raises an interesting point that we may be able to use this same idea  to enhance the curl terms, rather than nullifying them. This is indeed possible, although it would be challenging in the analytical framework of the paper. As we saw from our stability analysis, the architecture plays a crucial role in determining whether the curl terms will improve or impair learning performance, and a numerical treatment is unlikely to work without any theoretical guidance. A theoretical application to cortically defined structural connectivities will definitely be interesting but will require a new analytical framework (e.g., recurrent). We will add this point in the Discussion as both being a caveat of the work, and a potential future direction to find more robust curl terms.\n\n## 4. Expansion of Figure 5 \n\nWe fully agree that the empirical evidence presented in this work does not demonstrate the generalization of curl descent gains to more complex architectures or tasks. The primary contribution of this paper is to give a first treatment of non-gradient dynamics, in an analytically tractable setting.  Our results therefore show that within this setting, curl descent can enhance learning within a limited region of structural parameter space that can be analytically identified. To date, we lack analytical results that would guide the correct parameterization of more complex networks on real-world datasets. \n\nHowever, as suggested by reviewers mR3k and 2UnU, we will improve Figure 5 through additional empirical results regarding the effect of curl descent dynamics under other conditions, while still operating within the student-teacher setting described in our work. We conducted further numerical analyses on two recurring concerns from the reviewers’ feedback: the generalization of curl descent benefits to other architectural parameters and task complexity.\n\n### A. Architectural parameters\n\nWe conducted new simulations to investigate the effect of increasing the fraction of rule-flipped rule neurons in the readout layer for three compression ratios (0.1, 1 and 10). We found that the benefits of curl descent, as well as the range of rule-flipped neurons fraction improving learning, were more pronounced in networks with high compression. \n\nRegarding the introduction of rule-flipped rule neurons in the hidden layer, we did not find any structural parameters that lead to improved performance compared to gradient descent, consistent with our analytical results in linear networks (Figure 3a)\n\n### B. Modulating task complexity\n\nWe have added new simulations where we parametrically increase the complexity of the task by expanding the teacher’s initialization range. This is known to increase the complexity of the teacher representation and, consequently, the difficulty of the task the student must learn (Poole et al., 2016). These simulations reveal a threshold initialization range above which there is a sharp drop in convergence speed for both curl descent and gradient descent, as the dynamics in either case are stuck in a saddle. Above this threshold, both rules converge at the same (slow) speed. Below this threshold, curl descent converges faster than gradient descent."}, {"review_id": "sszrNr1qgo", "reviewer": "Reviewer_35Tb", "summary": "This paper introduces curl dynamics as an extension to standard gradient descent optimization, drawing inspiration from biological neural networks. Using a toy model, the authors offer clear intuition behind the proposed approach. They then analyze the stability of curl descent within a teacher-student framework using linear transformation networks, supporting their theoretical findings with numerical experiments.", "questions": "I noticed that the test error heat map (Fig. 4a) appears to differ significantly from the theoretical prediction (Fig. 2, bottom). In contrast, the max error heat map seems to align more closely with the theoretical results based on random matrix theory. I may be misunderstanding something, but the discrepancy between the numerical results and theoretical predictions does not seem to be fully addressed in the paper. Could the authors clarify this point?\n\nAdditionally, what is the origin of the U-shaped pattern observed in the test error heat map (Fig. 4a)? This U-shape appears even more pronounced when using the nonlinear tanh activation function, as shown in Supplementary Material (page 14, Fig. 3a). Is there an underlying theoretical explanation for this behavior?\n\nFor reference, the importance of curl dynamics in neural networks has been previously highlighted—for example, in Yan et al., Nonequilibrium landscape theory of neural networks, PNAS 110(45): E4185–E4194 (https://doi.org/10.1073/pnas.1310692110).", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strength:\nThe paper offers a solid theoretical foundation for curl descent, employing analytical techniques based on a toy model and random matrix theory to provide clear rationale and insight.\n\nWeakness:\nWhile the work serves as a strong proof of concept, it falls short by not demonstrating the practical effectiveness of curl descent in real-world deep learning scenarios with more realistic datasets.", "was_revised": true, "final_justification": "After reviewing the responses and related discussions, I have decided to maintain my original rating of 5 (accept).", "rebuttal": "We thank the reviewer for their constructive comments. We respond point-by-point below.\n\n## 1. Mismatch between test error heat map and theoretical prediction:\n\nIndeed, Figure 4a shows that the test error is low even when the analytical theory predicts that the solution manifold should become unstable. We were puzzled as well by this result, which led us to try initializing the weights very close to (but not directly on) the solution manifold, to test whether it was in fact numerically stable. \n\nWe quantified numerical stability using the maximum error along the trajectory when initialized near the solution manifold. Recall that the error is zero directly on the solution manifold. Therefore, a max error near zero means that the trajectory converged to the solution manifold (thus it is numerically stable). A large max error indicates that the trajectory climbed up the gradient, leading to a transient increase in the error. \n\nThese results are shown in Figure 4b, and precisely match the prediction from the analytical stability analysis. Together, Figure 4a and 4b demonstrate that the solution manifold does indeed lose stability as predicted by the theoretical results. However, this transition leads not to chaotic weight dynamics (as in Figure 3a), but rather towards other low-error solutions.\n\nWe will rework the text describing Figure 4 to better explain why this mismatch occurs, and why it is in fact consistent with the analytical theory. \n\n## 2. U-shaped pattern in the test error heat map:\n\nTo produce Figure 4a, we trained networks with a fixed 220 total number of neurons, varying the compression ratio $c$. We observed that the number of epochs to convergence increased with $c$ for both gradient descent and curl descent. In the case of curl descent, we noticed that as the curl terms were strengthened by increasing the fraction of rule flipped neurons, the transient learning dynamics lasted longer before settling into a low-error region. The red U-shape region of Figure 4a therefore corresponds to simulations that reached the $10^5$ epochs before the transient dynamics ended. Increasing the number of epochs effectively reduces the size of this U-shaped pattern. \n\nInterestingly, the bottom-right side of this U-shaped pattern (where $c=10$ and the fraction of rule-flipped  neurons = 0.1) delimits a region where the transient dynamics appear to be shorter. In this region, curl descent actually performs better than gradient descent in both the linear and the nonlinear cases (with the nonlinear case being the focus of figure 5). Unfortunately, we do not yet have a theoretical explanation for this phenomenon, but we will raise this interesting observation in the Discussion. \n\n## 3. Nonequilibrium landscape theory of neural networks\n\nWe thank the reviewer for this citation, which we originally missed. We will incorporate it into Section 6 – Related work.\n\n\n## Additional simulations for Figure 5\n\nWe fully agree that the empirical evidence presented in this work does not demonstrate the generalization of curl descent gains to more complex architectures or tasks. The primary contribution of this paper is to give a first treatment of non-gradient dynamics, in an analytically tractable setting. Our results therefore show that within this setting, curl descent can enhance learning within a limited region of structural parameter space that can be analytically identified. To date, we lack analytical results that would guide the correct parameterization of more complex networks on real-world datasets.\n\nHowever, as suggested by reviewers mR3k and 2UnU, we will improve Figure 5 through additional empirical results regarding the effect of curl descent dynamics under other conditions, while still operating within the student-teacher setting described in our work. We conducted further numerical analyses on two recurring concerns from the reviewers’ feedback: the generalization of curl descent benefits to other architectural parameters and task complexity.\n\n### A. Architectural parameters\n\nWe conducted new simulations to investigate the effect of increasing the fraction of rule-flipped rule neurons in the readout layer for three compression ratios (0.1, 1 and 10). We found that the benefits of curl descent, as well as the range of rule-flipped neurons fraction improving learning, were more pronounced in networks with high compression.\n\nRegarding the introduction of rule-flipped rule neurons in the hidden layer, we did not find any structural parameters that lead to improved performance compared to gradient descent, consistent with our analytical results in linear networks (Figure 3a)\n\n### B. Modulating task complexity\n\nWe have added new simulations where we parametrically increase the complexity of the task by expanding the teacher’s initialization range. This is known to increase the complexity of the teacher representation and, consequently, the difficulty of the task the student must learn (Poole et al., 2016). These simulations reveal a threshold initialization range above which there is a sharp drop in convergence speed for both curl descent and gradient descent, as the dynamics in either case are stuck in a saddle. Above this threshold, both rules converge at the same (slow) speed. Below this threshold, curl descent converges faster than gradient descent."}, {"review_id": "TEOTFQ8lbK", "reviewer": "Reviewer_mR3k", "summary": "In this paper, the authors demonstrate that learning is still possible in neural networks in which the sign of synaptic plasticity for some neurons is flipped. They develop a formal theory that characterizes the curl dynamics induced by such sign flips, and characterize the effects of sign flips on the fixed point structure of learning dynamics. Lastly, the authors show that under some conditions the presence of sign flips can, counterintuitively, be beneficial for optimization.", "questions": "Major concerns:\n\n1.\tIn section 2, it is unclear to me whether the curl descent analysis provided is useful, because unlike in the gradient descent framework in section 3, fixed points of the network are not well-characterized. This is relevant, because the Hebbian synaptic plasticity rule selected in Eq. 1 typically produces explosive weight dynamics if M_ij is positive, and vanishing dynamics if M_ij is negative. After all, if the plasticity rule is not stable or results in vanishing synaptic magnitudes, what is the value of this analysis?\n\n2.\tFor Figure 5, the authors have provided a demonstration that flipping the sign of plasticity for one readout weight improves performance significantly in a student teacher formulation. However, this is really not enough to be able to make much of any claim about the effects of curl descent on learning dynamics under other conditions. For instance, what happens as the number of flipped signs increases? What happens if signs are flipped in the hidden layer instead? What happens if the network is trained on an even moderately more complex task (e.g. MNIST), or if more complicated network architectures are used (e.g. convolutional)? Because this claim is counterintuitive (and potentially very cool), more empirical evidence is required to support the claim; while I accept the intuition that curl descent could help networks avoid local minima or slow points, making such a method function adequately at scale may require additional tricks (e.g. taking an annealing-style/learning rate scheduled approach to curl descent).\n\n3.\tWhile the authors seem to assume that sign flips in plasticity exclusively induce non-gradient curl terms in weight dynamics, this is not the only possibility, and alternatives should be discussed. I see four alternative possibilities. First, and least interesting, the functional form of ‘optimal’ synaptic plasticity within a normative framework can change quite radically based on the biophysics of a neuron and the ‘experimental’ conditions under which plasticity is characterized, meaning that cell-type specific plasticity could simply be a due to evolutionary corrections for differences in the biophysics of different neuron subtypes. Second, the sign of synaptic plasticity has been shown to vary considerably based on the position of a synapse relative to the soma in pyramidal neurons (Froemke et al., 2010; Sjostrom & Hausser, 2006); several other theoretical frameworks have argued that this switch in the sign of plasticity is actually fully consistent with approximate gradient descent learning (Richards & Lillicrap, 2019). Third, neuromodulators have been show to induce flips in the sign of synaptic plasticity, and many studies have attributed this effect to a natural consequence of reinforcement learning (Frémaux & Gerstner, 2016). And lastly, cell-type specific variations in plasticity could be due to the optimization of local auxiliary objectives that effectively place constraints on the optimization landscape for other non-local optimization processes (e.g. inhibitory plasticity could simply be serving the purposes of local homeostatic normalization of network activity). Recent studies have shown that network stabilization can emerge as a consequence of a wide diversity of different plasticity updates (e.g. Confavreux et al., 2025).\n\nMinor concerns:\n\n1.\tA recent paper (Shoji et al., 2024) has argued that a broad family of plasticity rules (even those with apparent curl) can be viewed as natural gradient descent if one chooses an appropriate timescale over which to average plasticity updates. While at face value this argument appears inconsistent with the conclusions of this paper, I actually think that they are fully consistent—either the plasticity covered in the current paper violates the constraints required for formal argument in the natural gradient study, or it may indeed be viewed as a form of natural gradient descent if averaged over an appropriate timescale. Regardless, the paper should be discussed because of their similar setup and differing conclusions; it would fit nicely into the ‘Natural gradients’ section.\n\n2.\tWhy can the first term in Eq. 2 be viewed as the negative gradient of an energy function? It is not immediately clear to me why the expression written can be viewed as an energy function (or what assumptions are being made to make it an energy function).\n\n3.\tIs it possible to generalize the mathematical analysis here to perturbations more general than sign flips? It seems to me like this analysis ought to be able to be generalized to a much broader family of diagonal perturbation matrices without much change to the underlying math.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is very clearly written, and quite a bit of effort is put into building the reader’s intuitions, which was much appreciated. It is also original, as far as I can tell. I think that this paper is very interesting, primarily because of the formal theory that the authors are able to develop to mathematically analyze learning dynamics in sign-flipped networks that learn via gradient-based signals—in my mind, this contribution alone is probably sufficient to warrant acceptance with some revisions, but other parts of the paper could be significantly improved. I think that the value of Section 2 is less clear, and that the authors could significantly improve the paper by clarifying a few points regarding that section (discussed below). I am also generally unconvinced by the results that curl descent can improve performance relative to gradient descent in more complicated networks or on more complicated tasks.", "was_revised": true, "final_justification": "The authors addressed my concerns, by providing both additional empirical evidence to support their claims and additional explanations that should increase the clarity of the paper.", "rebuttal": "We thank the reviewer for their constructive comments. We respond point-by-point below.\n\n# Major concerns\n\n## 1. Curl descent analysis of Hebbian/anti-Hebbian plasticity\n\nThe rationale for Section 2 is only to provide motivation for sign-diverse plasticity by showing how they can arise in biological neural networks, before moving to the more tractable teacher-student framework. We agree the rule in Eqn 1 would require additional terms to stabilize it, but this will not in general affect the curl terms. \n\nFor example, one common way to stabilize weight dynamics is by adding a leak term of $-\\gamma W$ to the plasticity rule. This is just the gradient of the Frobenius norm $-\\frac{\\gamma}{2} \\Vert W \\Vert_F^2$, and can be considered as a regularization added to the original objective defined by the rest of the plasticity rule (if such an objective exists). \n\nHowever, if we incorporate the term $-\\gamma W$ to the plasticity rule, this will not affect whether or not the entire rule can be written as a gradient. As we showed in Appendix A.2, the Jacobian of the original rule in Eqn 1 is not symmetric when $M$ is sign diverse. A symmetric Jacobian is necessary for the plasticity rule to be a gradient system. Since the leak term itself has a symmetric Jacobian, adding it to Eqn 1 cannot symmetrize the entire plasticity rule’s Jacobian. The same logic holds for any stabilization term that is itself gradient-based. \n\nTo avoid overburdening the reader, we had originally removed the leak term from Section 2 to focus only on the sign flips, but we acknowledge that this section may be confusing. In response to this reviewer’s feedback, as well as a suggestion from reviewer 2UnU, we will move the Hebbian/anti-Hebbian example to the appendix, incorporating the points above about the stabilization of the plasticity rule. We will keep only the EI network in the main text, and we will explicitly state that this section is only meant to motivate the sign flips before proceeding to the student-teacher framework that is used for the rest of the paper. Additionally, we will change the term ‘anti-Hebbian-like neurons’ for the more precise term 'rule-flipped neurons’. \n\n## 2. Additional simulations for Figure 5 \n\nAdditional simulations were conducted for figure 5, to which we added two panels, one regarding the generalization of curl descent benefits to a broader range of architectural parameters, and the other regarding task complexity. Due to limited space, we redirect the reviewer to the answer to reviewer 35Tb under the section “Additional simulations for Figure 5”.\n\nWe thank the reviewer for the excellent point about using tricks such as annealing to make curl descent more robust. We will add this to the Discussion.\n\n## 3. Alternatives to non-gradient curl terms\n\nWe fully agree that opposite-sign plasticity rules are not in themselves incompatible with gradient descent, and did not intend to suggest otherwise. Since gradient flow is the dominant normative framework for synaptic plasticity, we aimed to demonstrate how easily curl terms could arise through sign diverse plasticity rules. However, we acknowledge that a deeper discussion of these different possibilities would be useful.\n\nBelow we respond to each of the four possibilities.\n \n* In the first case, in networks with defined cell types, gradient descent can lead to diversity in the plasticity rules as a result of that heterogeneity. The reviewer mentions differences in intrinsic biophysical properties of individual neurons, which would predict a strong correlation between cell type and plasticity rule. Another example is diversity in structural connectivity. For example, Pehlevan and Chklovskii 2015 have shown that Hebbian plasticity in structured EI networks can implement gradient descent on a similarity matching objective. In Appendix A.4 we demonstrate that these networks are able to implement gradient flow by choosing a specific architecture that nullifies potential curl terms.\n\n* The second case of distally-dependent plasticity also brings up the importance that learning rules correlated with structural parameters (here distance to the soma) do have in biological neural networks. While we do not implement such structurally dependent learning rules in this work, we believe that this would be a very interesting direction to explore and will include this in the discussion.\n \n* Regarding sign-flips induced by neuromodulators, we thank the reviewer for pointing us to this body of work. This could provide a mechanism for biological networks to adaptively tune their curl terms in order to escape saddles and shallow minima. This could be a fascinating direction to explore in future work and could provide an alternative to reinforcement learning based framework in Fremeux and Gerstner.\n\n* Finally, it is true that different plasticity rules could be considered as separate objectives (homeostatic vs. error minimization) that may be locally implemented in different cell types. However, in curl descent, the plasticity rules are not diverse in their functional form, but rather have opposite signs. The rule-flipped neurons are ascending the gradient, and are therefore adversarial rather than homeostatic (see also our reply to 2UnU above). However, sign flipped rules could implement a homeostatic minimization if combined with cell-type-specific differences in connectivity profile or biophysical property, as in the first case above.\n \nWe will explicitly discuss these different possibilities where learning rules are flipped without causing curl terms in the Discussion section, along with the recommended citations.\n\n# Minor concerns\n\n## 1. Shoji et al., 2024\n\nWe thank the author for highlighting this work. We do cite this paper in Section 6 (Related work – Natural gradients), but we agree that it presents an interesting perspective that warrants more discussion. Shoji et al give a constructive proof that any learning rule which monotonically decreases a cost function after some time window implements a form of natural gradients, if we consider the integrated weight updates over this time window. In the case of curl descent, however, there may be regions of the parameter space where the learning dynamics are not monotonically decreasing, regardless of the time window considered. For example, in the toy model in Figure 1a, initializing within the heteroclinic orbits results in periodic weight dynamics, violating the core assumption of Shoji et al. In simulations in larger networks, we did not observe such non-decreasing loss curves (as long as only a minority of neurons in the readout layer were flipped), but we cannot rule out the existence of some weight initialization regimes that would lead to this. We will incorporate these points into the discussion on natural gradients.\n\n## 2. Energy function in Eqn 2\n\nHere we simply meant that this term by itself is the gradient of an objective function. Specifically, this term can be written as $\\mathbf{f}\\mathbf{f}^\\top \\odot M = -\\nabla_W \\mathcal{L}(W)$, where $\\mathcal{L}(W) = -\\text{Tr}((\\mathbf{f}\\mathbf{f}^\\top \\odot M) W^\\top)$. This loss is an inner product between $W$ and the matrix $(\\mathbf{f}\\mathbf{f}^\\top \\odot M) $. Therefore when $M = \\mathbf{1} \\mathbf{1}^\\top$ (no flipped synapses), this term tends to align the recurrent weights to the input covariance matrix. We’ll clarify this point in the text. \n\nWe also note that the term “energy function” may be misleading as it is often used to refer to gradient-based dynamics of the hidden unit activations (e.g., in Hopfield networks) rather than learning dynamics of $W$. To avoid misunderstanding, we will change this term simply to “objective function”. \n\n## 3. Generalization to diagonal matrices\n\nOur theoretical analysis could indeed be generalized to a broader range of diagonal perturbations. Assuming such general diagonal perturbations will not change the derivations leading to Eqns 13 and 14 in the main text. However, the subsequent random matrix theory derivations will have to be adapted. This would require using the Cauchy and Blue transforms (Eqns 89 and 90 of the supplementary materials) of a general diagonal matrix for perturbations of the hidden layer. Similarly, in the case of perturbations of the readout layer one could derive the support boundaries using the method of Rao & Edelman, 2006 (“The polynomial method of random matrices”), but here we expect a tedious calculation."}]

Author rebuttal:

Meta-review: The paper is motivated by biologically plausible training of neural networks using non-gradient based rules.  It introduces curl dynamics associated with sign flips and it develops an interesting theory, based on a teacher-student framework, that analyzes the stability of 
the learning dynamics. The authors provide experimental evidence showing that, in certain nonlinear architectures, their so called curl descent can even accelerate learning compared to gradient descent. 
 
As all reviewers agree, the paper introduces an original idea with a very clear motivation and exposition. Many attendees in the conference especially from the field of computational neuroscience will find this work very interesting.
Decision: Accept (spotlight)