Algebras of actions in an agent's representations of the world
This work addresses the problem of representing complex world transformations in AI agents, offering a theoretical extension beyond group symmetries, though it appears incremental as it builds on existing symmetry-based methods.
The paper proposes a framework to extract the algebra of world transformations from an agent's perspective, generalizing symmetry-based representation learning to handle any algebra, not just groups, and shows that disentangled sub-algebras can have independent equivariance conditions.
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.