# Convergence of energy-based learning in linear resistive networks

arXiv: https://arxiv.org/abs/2503.00349 · HTML: https://mlsanity.com/paper/2503.00349/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 62 references that are in the map.
- First posted on arXiv 1 Mar 2025.

## Builds on (most cited first, 14 of 14)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Roadmap to Neuromorphic Computing with Emerging Technologies](https://mlsanity.com/paper/2407.02353/) (2024)
- [Towards a Systems Theory of Algorithms](https://mlsanity.com/paper/2401.14029/) (2024)
- [The Analysis of Optimization Algorithms, A Dissipativity Approach](https://mlsanity.com/paper/2205.14264/) (2022)
- [Optimization Algorithm Synthesis based on Integral Quadratic Constraints: A Tutorial](https://mlsanity.com/paper/2306.00565/) (2023)
- [Dissipativity Tools for Convergence to Nash Equilibria in Population Games](https://mlsanity.com/paper/2005.03797/) (2020)
- [Monotone one-port circuits](https://mlsanity.com/paper/2111.15407/) (2021)
- [Circuit Analysis using Monotone+Skew Splitting](https://mlsanity.com/paper/2211.14010/) (2022)
- [Learning in memristive electrical circuits](https://mlsanity.com/paper/2409.15136/) (2024)

## Built on by (most cited first, 2 of 2)

- [Coercivity and Local Convergence of Physical Learning in Linear Circuits](https://mlsanity.com/paper/2606.15443/) (2026)
- [Circuit realization and hardware linearization of monotone operator equilibrium networks](https://mlsanity.com/paper/2509.13793/) (2025)

## Abstract (arXiv)

Energy-based learning algorithms are alternatives to backpropagation and are well-suited to distributed implementations in analog electronic devices. However, a rigorous theory of convergence is lacking. We make a first step in this direction by analysing a particular energybased learning algorithm, Contrastive Learning, applied to a network of linear adjustable resistors. It is shown that, in this setup, Contrastive Learning is equivalent to projected gradient descent on a convex function with Lipschitz continuous gradient, giving a guarantee of convergence of the algorithm for a range of stepsizes. This convergence result is then extended to a stochastic variant of Contrastive Learning.