# Finite-Nudge Equilibrium Propagation in Thermal Ensembles

arXiv: https://arxiv.org/abs/2511.22024 · HTML: https://mlsanity.com/paper/2511.22024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 43 references that are in the map.
- First posted on arXiv 27 Nov 2025.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Towards Biologically Plausible Deep Learning](https://mlsanity.com/paper/1502.04156/) (2015)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Generalization of Equilibrium Propagation to Vector Field Dynamics](https://mlsanity.com/paper/1808.04873/) (2018)
- [Equilibrium Propagation: the Quantum and the Thermal Cases](https://mlsanity.com/paper/2405.08467/) (2024)

## Abstract (arXiv)

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.