Lock-in EP: An In-Situ Training Algorithm for Oscillatory Hardware

Sowjanya Tammali; Wilkie Olin-Ammentorp

cs.LGOct 5, 2026

Lock-in EP: An In-Situ Training Algorithm for Oscillatory Hardware

Authors: Sowjanya Tammali, Wilkie Olin-Ammentorp

Organizations: Computer Science, Missouri University of Science and Technology, 500 West 15th Street, Rolla, 65409, MO, USA. · Mathematics & Computer Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Lemont, 60439, IL, USA.

Abstract

Analog hardware platforms offer the potential to reduce energy consumption over digital architectures, but in order to succeed, large-scale analog systems must also be able to operate with or recover from the variability of their components. Towards this goal, we derive and demonstrate the lock-in equilibrium propagation (LIEP) training method. LIEP provides local gradient information for each component in an oscillatory network without separate forward and backward sweeps, potentially allowing for in-situ learning capabilities on analog oscillatory hardware platforms. We demonstrate that LIEP can be used both for ab-initio training as well as recovering performance when pre-trained parameters are perturbed. We show that LIEP can be formulated as a three-factor update rule, and suggest that although the method is currently only validated on shallow networks, alternate architectures may allow it to extend to deep and large-scale networks addressing complex tasks.

Figures & tables

Panel of Figure 1 — Figure 1: Layers of neurons with states $z_{\ell}$ are connected via symmetric weights (left). The energy of each layer is defined as the non-alignment in phases between that layer’s current state and its input (Eq. 5 ). For intermediate layers (e.g. $z_{1}$ ), descending this derived energy function can be accomplished by following the “drive” function (Eq. 10 ) which is dependent on both the previous layer ( $z_{\ell-1}$ ) and next layer ( $z_{\ell+1}$ ). The phase term of the state $\theta_{j}$ aligns with the phase of the drive term $\theta_{D_{j}}$ , decreasing the layer’s energy over time. The desired outputs $y$ can influence the energy at each layer by contributing an output layer drive from a “virtual” layer ( $L+1$ ), normalized by the output dimensionality $d$ and modulated by $\beta$ .

Panel of Figure 2 — Figure 2: In standard neural networks, the state for a layer is reached by multiplying inputs by weights and applying a nonlinear “squashing” activation function $\rho$ (left). In EP networks, fixed states are reached by following the drive function $D$ at each layer (Eq. 10 ). In LIEP networks, each layer’s oscillatory state $z_{\ell}$ lies on the unit torus in the complex plane, where each point corresponds to set of phase values. To meet this prerequisite, the projection function $u$ maps the drive ( $D_{1}$ , $D_{2}$ , $D_{3}$ ) onto the torus (center). By utilizing a linear update of scaled drive steps (Eq. 12 ), the layer’s state converges to a fixed point on the unit torus over time (right).

Panel of Figure 3 — Figure 3: Utilizing the network’s energy function, a drive function combined with an appropriate update step allows the network to reach stable, fixed points given an input $x$ , desired output $y$ , and cost weighting term $\beta$ . When $\beta=0$ , the network’s “free” fixed point $z^{\circ}$ is generated. Varying $\beta$ influences the network’s energy, producing different fixed points when $y$ does not align with the network’s output. Varying $\beta$ from 0 to a positive value $\varepsilon$ is simple (a), but produces a biased estimate of the local gradient around the free fixed point. Adding a second sampling point at $-\varepsilon$ reduces the error of this estimate (b). Alternatively, we may modulate $\beta$ through time following a square-wave schedule to sweep it through the necessary values (c). We may further simplify the probe by replacing the square wave with a simple sinusoid which sweeps $\beta$ through time (d). This sinusoidal schedule replaces discrete “settling” stages, instead injecting the probe as a time-varying signal with a single harmonic component and allowing the local gradient at the free fixed point to be demodulated.

Panel of Figure 4 — Figure 4: Applying a time-varying cosine probe $\beta(t)$ continuously modulates the contribution of the cost function to the network’s energy function (a). As a result, the Hebbian energy term of each layer $H_{\ell}(t)$ varies through time as well (b). Removing the “DC” energy term of the network at its free fixed point $H^{\circ}_{\ell}$ and multiplying by the probe to “lock-in,” the resulting signal (c) can be averaged over multiple periods of the probe to estimate the gradient loss function w.r.t. network weights (Eq. 20 ) at each network layer (d). By sampling full periods, the explicit subtraction of the DC signal may also be dropped, as it averages to zero. Inspecting the spectrum of the energy $H_{\ell}$ , the “DC” free state sits at zero, with the cosine probe creating a clear sideband at its fundamental frequency $\omega_{p}$ and higher-order harmonics (e).

Panel of Figure 5 — Figure 5: We recapitulate the stages of our derivation, and highlight the relevant formulas and quantities passed from each stage to the next. By defining an energy function on the network $\Phi$ , we create an associated drive function $D$ which provides “steps” which can lower the energy of the network with inputs $x$ , target outputs $y$ , and weights $W$ (a). The per-layer drive $D_{\ell}$ is projected onto the unit torus in the complex plane, and a linear update step applies the drive to descend the energy function and “land” fixed network points $z^{\circ}$ on this torus (b). The contribution of the target outputs $y$ to the cost function, $\beta$ , is varied following a cosine schedule with amplitude $\varepsilon$ and frequency $\omega_{p}$ (c). The resulting time-varying per-layer Hebbian energy, $H_{\ell}(t)$ can then be demodulated to provide an estimate of the loss function gradient w.r.t. network weights at each layer (d). These gradients can then be applied to step the network through an outer loop which reduces the loss function over time.

Figure 6: LIEP is controlled by 3 major parameters: the frequency ( $\omega_{p}$ ) and amplitude ( $\varepsilon$ ) of the cosine probe, and the number of integration periods ( $T$ ) over which the lock-in signal is integrated. The quality of the gradients measured by their cosine similarity to BP-derived values is plotted across a grid where $\omega_{p}$ and $\varepsilon$ are swept, establishing the existence of an “adiabatic” basin. In this regime, the slow probe frequency and moderate probe amplitude allow the network to settle to steady states and produce high-quality gradient information (a). If the probe becomes too fast, approaching the rate at which the network descends to fixed points ( $\kappa$ ), the energy dynamics of the system can no longer produce this information, and the ability of the networks to train successfully collapses, showing high failure rates (b) and non-convergence of the loss function (d). Networks trained ab-initio with LIEP in the adiabatic region approach performance levels of standard BP on the same network architectures (c).

Experiment	Fig.	$\varepsilon$	$\omega_{p}$	$n_{\text{cyc}}$	$T_{\text{warm}}$	$T_{\text{free}}$	$\mathrm{d}t$	$T$	Train
Adiabatic grid	6 a,b	swept	swept	swept	1	200	0.5	varies	—
Basin training	6 c,d	0.003–0.03	0.01–0.10	4	1	200	0.5	varies	60k
Impairment sweep	7 a–d	0.03	0.02	4	2	200	0.5	2512	60k
Three-factor identity	8	0.05	0.05	4	2	100	0.1	5028	—

Table 1: Lock-in probe settings behind each figure.

T

is the demodulation window in integrator steps,

n_{\text{cyc}}

periods of the probe; a further

T_{\text{warm}}

periods are discarded before accumulation begins. All runs use

784\to 256\to 64

at initial weight scale

0.4

. With

\kappa=0.080

, the

\omega_{p}=0.02

used for the impairment studies is

\omega_{p}/\kappa=0.25

— point B of Fig. 6 , well inside the adiabatic basin and about a factor of two below the fitted boundary. Two settings are not shared across Fig. 7 : its panel (b) uses two warm-up periods and the full

60

k training set, while the paired comparison in (c)–(d) uses one and

10

k. The paired design is internally controlled, but accuracies are not directly comparable between the two.

Figure 7: Rather than competing directly with standard backpropagation (BP) for large-scale training of digital networks, LIEP may be best applied as an analog-native training algorithm for correcting device-level distortions as pretrained networks are deployed onto devices. Many analog devices, such as resistive memories, have limited precision which causes the desired weights to be mapped to a variety of states. This mapping may follow a symmetric Gaussian curve (a), a log-normal scaling (b), or map weights to entirely broken “open” devices (c) or “open/shorted” devices (d). Each of these cases is applied at varying levels of severity, and BP on the entire network or only the readout layer is applied as a control correction. These methods are compared against the experimental LIEP. For many levels of disruption, LIEP can provide recovery approaching that of full-network BP. All accuracies plotted are the median of 8 samples.

Figure 8: LIEP may be reformulated as a local, 3-factor learning rule. We begin by revisiting the final, matrix form of the loss-reducing weight update step (a). The quantities required for this update can be separated into locally-available energy information at each synapse (b) which measures the phase alignment of the pre- and post-synaptic neurons. This alignment is compared to the current state of the global probe (c), which allows the demodulated weight update to be accumulated and integrated into an eligibility trace (d) that can be used to update each weight locally.

Tensor	Components	$1-\cos$	median	max
		Worst	Relative error
$W_{1}$	200 704	$4\times 10^{-6}$	$1.2\times 10^{-3}$	$2.8\times 10^{-3}$
$W_{2}$	16 384	$3\times 10^{-7}$	$3.8\times 10^{-4}$	$5.4\times 10^{-4}$

Table 2: The numerical results of the three-factor formulation of LIEP are compared against the original, network-level implementation. The same shallow MLP network used previously (Sec. 3.1 ) is optimized using the original LIEP formulation (Sec. 2.2 ) and its 3-factor version derived here. For all components in the network, all match to relative errors less than several parts per thousand.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Nudge $\beta(t)$	component at $+\omega_{p}$	component at $-\omega_{p}$	what is extracted
$\pm\varepsilon$ (static)	$\pm\varepsilon(a+b)$ , at DC		$a+b$ , from two or three settles
$\varepsilon e^{\mathrm{i}\omega_{p}t}$ (complex)	$\varepsilon a$	$\varepsilon b$	$a$ only, if demodulated at $+\omega_{p}$
$\varepsilon\cos(\omega_{p}t)$ (real)	$\tfrac{\varepsilon}{2}(a+b)$	$\tfrac{\varepsilon}{2}(a+b)$	$a+b$ , from one settle

Appendix

Table A.1: Response of the equilibrium to three nudging protocols, to first order in the probe amplitude.

Quantity	Value	Quantity	Value
$\rho_{1}=\\|W_{1}\\|_{2}$	$6.02$	$\varrho(G)$ from min drive	$359$
$\rho_{2}=\\|W_{2}\\|_{2}$	$2.10$	from $1\%$ quantile	$4.90$
$\mu_{1}$ (min drive)	$0.017$	from $10\%$ quantile	$2.13$
$\mu_{2}$ (min drive)	$0.0020$	from $25\%$ quantile	$1.53$
median drive, layer $1$	$4.21$	from median drive	$1.09$
median drive, layer $2$	$0.877$	required	$<1$

Appendix

Table B.1: Measured on the trained

784\to 256\to 64

network,

500

test samples.

\varrho(G)

is reported from the drive minimum ( B.13 ) and from four quantiles of the drive distribution; the median row is the most generous reading any refinement of ( B.17 ) could achieve.

Architecture	$L$	$\rho_{L}/\rho_{1}$	$\varrho(G)$	out dev	BP	EP	drop	bound
$784\to 256\to 64$	$2$	$0.350$	$1.095$	$0.285$	$0.882$	$0.870$	$+0.012$	$0.162$
$784\to 256\to 256\to 64$	$3$	$0.308$	$1.838$	$0.774$	$0.880$	$0.828$	$+0.052$	$0.782$
$784\to 256\to 256\to 256\to 64$	$4$	$0.277$	$2.118$	$0.975$	$0.876$	$0.742$	$+0.134$	$0.876$
$784\to 64\to 64$	$2$	$0.576$	$1.331$	$0.985$	$0.868$	$0.818$	$+0.050$	$0.846$
$784\to 128\to 64$	$2$	$0.417$	$1.186$	$0.378$	$0.862$	$0.864$	$-0.002$	$0.278$
$784\to 512\to 64$	$2$	$0.305$	$1.078$	$0.262$	$0.878$	$0.872$	$+0.006$	$0.136$

Appendix

Table B.2: Depth and width sweep, seed

42

500

test images, five epochs of Adam per configuration. “out dev” is the mean relative deviation

\hat{\delta}_{L}

at the output layer, against a trivial maximum of

2

; “bound” is Proposition B.7 evaluated per sample. Every row satisfies bound

\geq

drop.

Explore similar work

Coercivity and Local Convergence of Physical Learning in Linear Circuits
Jun 13, 2026Joshua A. McGinnis, Xinbo Li, Yoichiro MoriUncoupled Learning Convergence
Department of Mathematics, University of Pennsylvania, Philadelphia.
Dynamic Symmetric Point Tracking: Tackling Non-ideal Reference in Analog In-memory Training
Feb 24, 2026Quan Xiao, Jindan Li, Zhaoxian Wu +2In-Memory Computing Reram Crossbar
Department of Electrical and Computer Engineering, Cornell University, New York, NY · IBM T. J. Watson Research Center, Yorktown Heights, NY · Rensselaer Polytechnic Institute, Troy, NY
Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learning
Jun 8, 2026Chen-Rui Fan, Bo Lu, Xing-Yu Wu +2Hopfield Networks Phase Space
School of Artificial Intelligence, Beijing Normal University, Beijing 100875, China · Laboratory for Advanced Computing and Intelligence Engineering, Information Engineering University, Zhengzhou 450001, China · School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

Jun 13, 2026, math.OCJ/K move · Enter open · S save

Coercivity and Local Convergence of Physical Learning in Linear Circuits

Joshua A. McGinnis, Xinbo Li, Yoichiro Mori

Department of Mathematics, University of Pennsylvania, Philadelphia.

Uncoupled Learning Convergence Circuits Optical Networks Emotional Manipulation

Open paper AskPDF arXiv

Physical learning methods train physical networks to perform computational tasks using only local update rules, exploiting the physics of the system to handle the global transfer of information. We provide the first local convergence analysis of three such methods -- Equilibrium Propagation (EP), Coupled Learning (CL), and a new method we call Adjoint Coupled Learning (AL) -- for linear circuits, in the limit of small-nudging for both discrete and continuous time. EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. Assuming the existence of a solution, we identify a coercivity condition, expressed as a rank condition on a matrix built from the network's incidence structure, under which the training loss decays exponentially and the parameters converge to the solution manifold. We show that coercivity can fail by exhibiting a kite circuit in which a symmetry causes the coercivity constant to degenerate on the solution manifold, but prove using Sard's theorem that such degeneracies are non-generic: coercivity holds at every point of the solution manifold for almost every choice of desired output.

Abstract

Explore similar work

Coercivity and Local Convergence of Physical Learning in Linear Circuits

Dynamic Symmetric Point Tracking: Tackling Non-ideal Reference in Analog In-memory Training

Hybridizing Equilibrium Propagation with Ising Machines for Efficient Energy-Based Learning