Finite-Nudge Equilibrium Propagation in Thermal Ensembles

We liberate Equilibrium Propagation (EP) from the limit of infinitesimal perturbations by establishing a finite-nudge foundation for local credit assignment. By modeling network states as Gibbs-Boltzmann distributions rather than deterministic points, we prove that the gradient of the difference in Helmholtz free energy between a nudged and free phase is exactly the difference in expected local energy derivatives. This validates the classic Contrastive Hebbian Learning update as an exact gradient estimator for arbitrary finite nudging, requiring neither infinitesimal approximations nor convexity. In the zero-temperature limit, we prove that the same identity reduces to the deterministic contrastive rule around any local energy basin without assuming a unique global minimum, and a subsequent small-nudge limit recovers traditional EP. Finally, we derive an equivalent representation of the same gradient as an integral of the loss--energy covariance over nudging strength, which generalizes infinitesimal EP to strong error signals that its small-nudge approximation cannot support. Numerical experiments corroborate that finite nudging provides a practical signal-to-noise advantage over infinitesimal methods during training.