# New Insights on Learning Rules for Hopfield Networks: Memory and Objective Function Minimisation

arXiv: https://arxiv.org/abs/2010.01472 · HTML: https://mlsanity.com/paper/2010.01472/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2020.

## Built on by (most cited first, 5 of 5)

- [Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity](https://mlsanity.com/paper/2510.02765/) (2025)
- [Hardware-Adaptive and Superlinear-Capacity Memristor-based Associative Memory](https://mlsanity.com/paper/2505.12960/) (2025)
- [Novel Complex-Valued Hopfield Neural Networks with Phase and Magnitude Quantization](https://mlsanity.com/paper/2507.00461/) (2025)
- [Redundancy Maximization as a Principle of Associative Memory Learning in Hopfield Networks](https://mlsanity.com/paper/2511.02584/) (2025)
- [Implicit Bias and Invariance: How Hopfield Networks Efficiently Learn Graph Orbits](https://mlsanity.com/paper/2512.14338/) (2025)

## Abstract (arXiv)

Hopfield neural networks are a possible basis for modelling associative memory in living organisms. After summarising previous studies in the field, we take a new look at learning rules, exhibiting them as descent-type algorithms for various cost functions. We also propose several new cost functions suitable for learning. We discuss the role of biases (the external inputs) in the learning process in Hopfield networks. Furthermore, we apply Newtons method for learning memories, and experimentally compare the performances of various learning rules. Finally, to add to the debate whether allowing connections of a neuron to itself enhances memory capacity, we numerically investigate the effects of self coupling. Keywords: Hopfield Networks, associative memory, content addressable memory, learning rules, gradient descent, attractor networks