Unifying Causal Inference and Reinforcement Learning using Higher-Order Category Theory

TL;DR

This paper introduces a unified mathematical framework using higher-order category theory to connect causal inference and reinforcement learning, enabling advanced structure discovery through simplicial objects and horn-filling problems.

Contribution

It develops a novel categorical formalism that unifies causal model discovery and predictive state representations in reinforcement learning using simplicial objects and horn-filling techniques.

Findings

Unified formalism for causal and RL models

Use of simplicial objects to represent structures

Solution approach via horn-filling in higher categories

Abstract

We present a unified formalism for structure discovery of causal models and predictive state representation (PSR) models in reinforcement learning (RL) using higher-order category theory. Specifically, we model structure discovery in both settings using simplicial objects, contravariant functors from the category of ordinal numbers into any category. Fragments of causal models that are equivalent under conditional independence -- defined as causal horns -- as well as subsequences of potential tests in a predictive state representation -- defined as predictive horns -- are both special cases of horns of a simplicial object, subsets resulting from the removal of the interior and the face opposite a particular vertex. Latent structure discovery in both settings involve the same fundamental mathematical problem of finding extensions of horns of simplicial objects through solving lifting problems in commutative diagrams, and exploiting weak homotopies that define higher-order symmetries. Solutions to the problem of filling "inner" vs "outer" horns leads to various notions of higher-order categories, including weak Kan complexes and quasicategories. We define the abstract problem of structure discovery in both settings in terms of adjoint functors between the category of universal causal models or universal decision models and its simplicial object representation.