🤖 AI Summary
This work addresses the challenge of efficiently training diffusion models on analog hardware by proposing a training method that requires no external digital accelerators while preserving a low-rank coupling structure. By integrating Symmetric Equilibrium Propagation into a bilinear energy-based model, the authors devise a local learning rule that relies solely on readout operations, reducing gradient bias from first- to second-order within a limited relaxation time. This substantially improves alignment between the update direction and the ideal gradient, enhancing training stability. Coupled with a physically grounded model of unit energy consumption, the approach achieves 10³–10⁴ times lower energy per training step compared to an equivalent GPU baseline, all while maintaining competitive model performance.
📝 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.