# Minimal Neural Network Models for Permutation Invariant Agents

arXiv: https://arxiv.org/abs/2205.07868 · HTML: https://mlsanity.com/paper/2205.07868/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 64 references that are in the map.
- First posted on arXiv 12 May 2022.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [On the Binding Problem in Artificial Neural Networks](https://mlsanity.com/paper/2012.05208/) (2020)
- [MetaMorph: Learning Universal Controllers with Transformers](https://mlsanity.com/paper/2203.11931/) (2022)
- [The Sensory Neuron as a Transformer: Permutation-Invariant Neural Networks for Reinforcement Learning](https://mlsanity.com/paper/2109.02869/) (2021)
- [Bootstrapping Conditional GANs for Video Game Level Generation](https://mlsanity.com/paper/1910.01603/) (2019)
- [Born to Learn: the Inspiration, Progress, and Future of Evolved Plastic Artificial Neural Networks](https://mlsanity.com/paper/1703.10371/) (2017)
- [Introducing Symmetries to Black Box Meta Reinforcement Learning](https://mlsanity.com/paper/2109.10781/) (2021)
- [Interactive Evolution and Exploration Within Latent Level-Design Space of Generative Adversarial Networks](https://mlsanity.com/paper/2004.00151/) (2020)
- [Evolving and Merging Hebbian Learning Rules: Increasing Generalization by Decreasing the Number of Rules](https://mlsanity.com/paper/2104.07959/) (2021)
- [Evolving Plasticity for Autonomous Learning under Changing Environmental Conditions](https://mlsanity.com/paper/1904.01709/) (2019)
- [Testing the Genomic Bottleneck Hypothesis in Hebbian Meta-Learning](https://mlsanity.com/paper/2011.06811/) (2020)

## Built on by (most cited first, 4 of 4)

- [General-Purpose In-Context Learning by Meta-Learning Transformers](https://mlsanity.com/paper/2212.04458/) (2022)
- [Structurally Flexible Neural Networks: Evolving the Building Blocks for General Agents](https://mlsanity.com/paper/2404.15193/) (2024)
- [LayerShuffle: Enhancing Robustness in Vision Transformers by Randomizing Layer Execution Order](https://mlsanity.com/paper/2407.04513/) (2024)
- [The Generalist Brain Module: Module Repetition in Neural Networks in Light of the Minicolumn Hypothesis](https://mlsanity.com/paper/2507.12473/) (2025)

## Abstract (arXiv)

Organisms in nature have evolved to exhibit flexibility in face of changes to the environment and/or to themselves. Artificial neural networks (ANNs) have proven useful for controlling of artificial agents acting in environments. However, most ANN models used for reinforcement learning-type tasks have a rigid structure that does not allow for varying input sizes. Further, they fail catastrophically if inputs are presented in an ordering unseen during optimization. We find that these two ANN inflexibilities can be mitigated and their solutions are simple and highly related. For permutation invariance, no optimized parameters can be tied to a specific index of the input elements. For size invariance, inputs must be projected onto a common space that does not grow with the number of projections. Based on these restrictions, we construct a conceptually simple model that exhibit flexibility most ANNs lack. We demonstrate the model's properties on multiple control problems, and show that it can cope with even very rapid permutations of input indices, as well as changes in input size. Ablation studies show that is possible to achieve these properties with simple feedforward structures, but that it is much easier to optimize recurrent structures.