Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training

TL;DR

This paper introduces symmetric equilibrium propagation for thermodynamic diffusion training, enabling energy-efficient, local, readout-only learning rules that can be physically realized without digital gradient routing.

Contribution

It demonstrates that equilibrium propagation can be applied directly to a bilinear substrate for unbiased gradient estimation, improving bias properties with symmetric nudging and achieving significant energy savings.

Findings

Bias bound controlled by substrate properties

Symmetric nudging reduces bias from O(β) to O(β^2)

Achieves 10^3-10^4× energy advantage over GPU baseline

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}(\beta) $ to $ \mathcal{O}(\beta^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.