The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

Mani Rash Ahmadi

cs.LGApr 11, 2026

The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

Authors: Mani Rash Ahmadi

Organizations: Calibur Labs

Abstract

We prove that in a coupled Kuramoto oscillator network at stable equilibrium, the physical phase displacement under weak output nudging is the gradient of the loss with respect to natural frequencies, with equality as the nudging strength beta tends to zero. Prior oscillator equilibrium propagation work explicitly set aside natural frequency as a learnable parameter; we show that on sparse layered architectures, frequency learning outperforms coupling-weight learning among converged seeds (96.0% vs. 83.3% at matched parameter counts, p = 1.8e-12). The approximately 50% convergence failure rate under random initialization is a loss-landscape property, not a gradient error; topology-aware spectral seeding eliminates it in all settings tested (46/100 to 100/100 seeds on the primary task; 50/50 on a second task, K-only training, and a larger architecture).

Figures & tables

Figure 1: The two-phase gradient readout protocol. (a) Solve for the free equilibrium

\bm{\theta}^{*}

. (b) Add a weak nudging force

-\beta(\theta_{i}-\theta_{i}^{\mathrm{tgt}})

at output oscillators and re-solve for

\bm{\theta}^{\beta}

. (c) The per-node phase displacement

(\theta_{k}^{\beta}-\theta_{k}^{*})/\beta

approximates the negative gradient

-\partial L/\partial\omega_{k}^{c}

at finite

\beta

, with equality in the limit

\beta\to 0

(Theorem 6 ). Blue: input nodes (frequencies set by data). Gray: hidden nodes. Amber: output nodes. Dashed line in (b): target phase.

$N$	Freq. params	Cosine (TP vs FD)	Cosine (AG vs TP)	Eq. residual
6	5	1.000000	1.000000	$8.4\times 10^{-14}$
10	9	1.000000	1.000000	$2.2\times 10^{-16}$
15	14	1.000000	1.000000	$3.4\times 10^{-14}$
20	19	1.000000	1.000000	$3.0\times 10^{-15}$
30	29	1.000000	1.000000	$1.1\times 10^{-15}$
50	49	1.000000	1.000000	$1.1\times 10^{-16}$

Table 1: Phase-gradient identity verification. Cosine similarity between gradient vectors computed by the scipy-based two-phase (TP) and finite-difference (FD) methods, and by the computationally independent PyTorch autograd (AG) method.

Figure 2: Gradient identity verification. (a) Per-node scatter of the two-phase gradient vs. the analytical gradient at

N=15

\beta=10^{-3}

: all points fall on

y=x

(cosine similarity

=1.000000

). (b) Equilibrium residual across network sizes from

N=6

N=200

; all residuals are at or below machine

\epsilon

. The cosine similarity is

1.000000

at every scale.

Coupling asymmetry	Cosine (TP vs FD)
0%	$1.000000\pm 0.000000$
5%	$0.999766\pm 0.000230$
10%	$0.999026\pm 0.001001$
20%	$0.995767\pm 0.004742$
50%	$0.965372\pm 0.046598$

Table 2: Asymmetry robustness (

N=15

, 10 seeds per level).

Task	Params	Conv. rate	Conv. test acc.	All-seed test
/a/ vs /i/	$\omega$ only	47% (47/100)	$94.3\%$	$48.9\%$
	$K$ only	48% (48/100)	$82.3\%$	$48.7\%$
	Both	48% (48/100)	$92.2\%$	$49.4\%$

Table 3: Parameter ablation (100 seeds, 150 epochs, lr = 0.001). Convergence defined by final-epoch training accuracy

>60\%

; reported accuracy is final-epoch test accuracy. Features normalized using training-set statistics only.

Figure 3: Parameter ablation on /a/ vs /i/ (100 seeds, sparse layered 2+5+2 topology). (a) Violin plots of all 100 seeds showing the bimodal distribution: 47/100

\omega

-only seeds and 48/100

K

-only seeds training-converge (final-epoch training accuracy above 60%; convergence rate difference not significant, Fisher

p=1.0

). Purple: joint (

\omega+K

) training. (b) Training-converged seeds only:

\omega

-only achieves 94.3% vs.

K

-only 82.3% (

p=1.5\times 10^{-10}

, Welch’s

t

-test).

Condition	Params	Conv. rate	Conv. test acc.
$\omega$ -only	7	47% (47/100)	96.0%
$K$ -matched	7	46% (46/100)	83.3%
$K$ -full	24	47% (47/100)	83.0%

Table 4: Parameter-matched ablation (100 seeds, 200 epochs, lr = 0.001). Convergence defined by final-epoch training accuracy

>60\%

. At equal parameter counts (7 vs. 7),

\omega

-only outperforms

K

-matched by 12.7 percentage points. Tripling

K

’s parameter budget does not close the gap.

$N$	$\omega$ params	$K$ params	$\omega$ conv. acc.	$K$ conv. acc.	Gap
7	5	14	93.6%	83.9%	+9.7
9	7	24	95.5%	81.8%	+13.7
11	9	34	89.6%	82.5%	+7.1
14	12	49	90.5%	82.9%	+7.6
19	17	74	88.2%	86.7%	+1.5

Table 5: Architecture sweep (50 seeds per size, 200 epochs).

\omega

-only outperforms

K

-only at every network size despite having 3–4

\times

fewer parameters.

	Converged (18/40)	Failed (22/40)
Init cond( $\tilde{J}$ )	$15.2\pm 1.9$	$15.4\pm 1.8$
Final cond( $\tilde{J}$ )	$14.5\pm 1.6$	$14.7\pm 1.5$
Skip rate	0.0	0.0

Table 6: Convergence diagnosis (40 seeds, 150 epochs). The gradient identity holds for all seeds, including those that fail to learn.

Figure 4: Convergence failure is a landscape problem, not a gradient problem. A converging seed (green) and a failing seed (amber) show different accuracy trajectories but indistinguishable diagnostics: the Jacobian condition number stays

\approx 15

and the skip rate is zero for both seeds.

Initialization	Success rate	Success accuracy	Overall
Random	46/100 (46%)	$96.8\%\pm 5.1\%$	48.5%
Spectral seed	100/100 (100%)	$97.6\%\pm 2.8\%$	97.6%

Table 7: Spectral seeding vs. random initialization (100 seeds,

\omega

-only, 200 epochs). Success defined as final test accuracy exceeding 60%.

Figure 5: Spectral seeding eliminates the random-initialization basin failure (100 seeds,

\omega

-only, 200 epochs). (a) Random initialization produces a bimodal distribution: 46/100 seeds exceed 60% test accuracy. Spectral seeding places all 100 seeds above 90%. (b) Per-seed accuracy change: 54 seeds rescued (green), 0 lost.

Setting	Init	Success	Acc.
/o/ vs /u/, $\omega$ -only, $2{+}5{+}2$	Random	19/50	76.5%
	Spectral	50/50	77.9%
/a/ vs /i/, $K$ -only, $2{+}5{+}2$	Random	22/50	81.4%
	Spectral	50/50	84.6%
/a/ vs /i/, $\omega$ -only, $2{+}8{+}2$	Random	21/50	91.4%
	Spectral	50/50	92.7%

Table 8: Spectral seeding generalization (50 seeds per setting, 200 epochs).

Equilibrium Propagation

Equilibrium propagation trains energy-based neural networks by estimating gradients from changes between free and weakly nudged equilibrium states.

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Mean--Fluctuation Dynamics at the Edge of Stability

Antonin Chodron de Courcel

Ecole Normale Sup´erieure, CNRS, 45 rue d’Ulm, 75005 Paris, France

Edge of Stability Gradient Descent Dynamics Neural Network Stability

Open paper AskPDF arXiv

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the trajectory, which has been linked to better generalization performance. We introduce the mean--fluctuation dynamics, a tractable continuous-time model coupling the window-averaged trajectory to its fluctuation covariance. Among our contributions, we rigorously derive this model from gradient descent in a sharp-valley framework, characterize its stationary states and their linear stability, and establish precise connections with other effective dynamics. Numerical experiments illustrate these predictions and their finite-time limitations. We also study our model in the overparametrized regime of wide two-layer networks at a fixed learning rate, where we rigorously derive a kinetic equation describing weights and their fluctuations as a Wasserstein-2 gradient flow, for which we prove well-posedness, a mean-field limit, and conditional convergence results.

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