Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training

• URL: http://arxiv.org/abs/2604.23806v1
• Date: Sun, 26 Apr 2026 17:05:05 GMT
• Title: Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training
• Abstract summary: We show that a bilinearly-coupled analog substrate can physically realize dynamics at a projected three-to-four orders of magnitude energy advantage over digital inference.<n>We resolve whether the emphtraining loop can be closed on the same substrate without routing gradients through an external digital accelerator.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The reverse process in score-based diffusion models is formally equivalent to overdamped Langevin dynamics in a time-dependent energy landscape. In our prior work we showed that a bilinearly-coupled analog substrate can physically realize this dynamics at a projected three-to-four orders of magnitude energy advantage over digital inference by replacing dense skip connections with low-rank inter-module couplings. Whether the \emph{training} loop can be closed on the same substrate -- without routing gradients through an external digital accelerator -- has remained open. We resolve this affirmatively: Equilibrium Propagation applied directly to the bilinear energy yields an unbiased estimator of the denoising score-matching gradient in the zero-nudge limit. For finite nudging we derive a sharp bias bound controlled solely by substrate stiffness, local curvature, and the norm of the loss-gradient signal, with a bilinear-specific corollary showing that one dominant bias term vanishes identically for coupling-parameter updates. Symmetric nudging further upgrades the leading bias from $ \mathcal{O}(β) $ to $ \mathcal{O}(β^2) $ at negligible extra cost. Under realistic finite-relaxation budgets this upgrade is essential, as one-sided EqProp produces anti-correlated gradients while symmetric EqProp yields well-aligned updates. Bias-variance analysis determines the optimal operating point, and end-to-end physical-unit accounting projects a $ 10^3$-$10^4\times $ energy advantage per training step over a matched GPU baseline. Symmetric bilinear EqProp is the first local, readout-only training rule that preserves the low-rank coupling enabling scalable thermodynamic diffusion models.

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