# Permutation-Invariant Set Autoencoders with Fixed-Size Embeddings for Multi-Agent Learning

## TL;DR

This paper introduces PISA, a permutation-invariant set autoencoder that produces fixed-size, accurate, and similarity-preserving embeddings, improving multi-agent learning and graph neural network applications.

## Contribution

The paper presents PISA, a novel permutation-invariant autoencoder with fixed-size embeddings that outperforms existing methods in reconstruction accuracy and supports element insertion/removal.

## Key findings

- PISA achieves significantly lower reconstruction error than baselines.
- PISA provides a similarity-preserving latent space.
- Using PISA, a new GNN architecture enables agents to communicate for full system observability.

## Abstract

The problem of permutation-invariant learning over set representations is particularly relevant in the field of multi-agent systems -- a few potential applications include unsupervised training of aggregation functions in graph neural networks (GNNs), neural cellular automata on graphs, and prediction of scenes with multiple objects. Yet existing approaches to set encoding and decoding tasks present a host of issues, including non-permutation-invariance, fixed-length outputs, reliance on iterative methods, non-deterministic outputs, computationally expensive loss functions, and poor reconstruction accuracy. In this paper we introduce a Permutation-Invariant Set Autoencoder (PISA), which tackles these problems and produces encodings with significantly lower reconstruction error than existing baselines. PISA also provides other desirable properties, including a similarity-preserving latent space, and the ability to insert or remove elements from the encoding. After evaluating PISA against baseline methods, we demonstrate its usefulness in a multi-agent application. Using PISA as a subcomponent, we introduce a novel GNN architecture which serves as a generalised communication scheme, allowing agents to use communication to gain full observability of a system.