Algebras of actions in an agent's representations of the world

TL;DR

This paper introduces a framework for extracting and classifying the algebraic structures of world transformations in an agent's representations, extending symmetry-based representation learning to more general algebraic cases.

Contribution

It generalizes the symmetry-based disentangled representation learning formalism to include arbitrary algebras of transformations, not just groups, and develops methods to classify and analyze these algebras.

Findings

Extracted algebras of world transformations in simple RL scenarios.

Classified these algebras based on their properties.

Extended equivariance and disentangling concepts to general algebras.

Abstract

In this paper, we propose a framework to extract the algebra of the transformations of worlds from the perspective of an agent. As a starting point, we use our framework to reproduce the symmetry-based representations from the symmetry-based disentangled representation learning (SBDRL) formalism proposed by [1]; only the algebra of transformations of worlds that form groups can be described using symmetry-based representations. We then study the algebras of the transformations of worlds with features that occur in simple reinforcement learning scenarios. Using computational methods, that we developed, we extract the algebras of the transformations of these worlds and classify them according to their properties. Finally, we generalise two important results of SBDRL - the equivariance condition and the disentangling definition - from only working with symmetry-based representations to working with representations capturing the transformation properties of worlds with transformations for any algebra. Finally, we combine our generalised equivariance condition and our generalised disentangling definition to show that disentangled sub-algebras can each have their own individual equivariance conditions, which can be treated independently.