Causal Optimal Transport of Abstractions
(2312.08107)Abstract
Causal abstraction (CA) theory establishes formal criteria for relating multiple structural causal models (SCMs) at different levels of granularity by defining maps between them. These maps have significant relevance for real-world challenges such as synthesizing causal evidence from multiple experimental environments, learning causally consistent representations at different resolutions, and linking interventions across multiple SCMs. In this work, we propose COTA, the first method to learn abstraction maps from observational and interventional data without assuming complete knowledge of the underlying SCMs. In particular, we introduce a multi-marginal Optimal Transport (OT) formulation that enforces do-calculus causal constraints, together with a cost function that relies on interventional information. We extensively evaluate COTA on synthetic and real world problems, and showcase its advantages over non-causal, independent and aggregated COTA formulations. Finally, we demonstrate the efficiency of our method as a data augmentation tool by comparing it against the state-of-the-art CA learning framework, which assumes fully specified SCMs, on a real-world downstream task.
Overview
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The paper introduces Causal Optimal Transport of Abstractions (COTA), a method that uses a multi-marginal Optimal Transport framework to align structural causal models (SCMs) of different resolutions.
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COTA does not require complete knowledge of the SCMs involved and is based on both observational and interventional data.
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It involves a multi-marginal formulation with causal constraints and provides a specific algorithmic approach for implementation.
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Tested on both synthetic and real-world data, COTA outperforms traditional non-causal methods and demonstrates practical efficacy in data augmentation tasks.
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The paper establishes a proof of convexity for the optimization problem, ensuring that the results are the best possible under the given constraints.
Introduction to Causal Optimal Transport of Abstractions
Causal abstraction (CA) works to align structural causal models (SCMs) that operate at varied levels of detail, creating maps between them. These maps are crucial for several real-world problems, such as synthesizing causal evidence across different environments or learning about causal relationships at diverse resolutions. This paper introduces a novel method handling these maps without assuming complete knowledge of the SCMs involved, using a technique based on multi-marginal Optimal Transport (OT) theory.
Background: Causality and Optimal Transport
In causality, SCMs are leveraged to understand the relationships and interactions between variables within a system. Each SCM contains variables, functions, and a probability distribution over these variables. By manipulating these variables through interventions, one can observe different outcomes and thus infer causal connections.
Causal abstractions allow for the synthesis of information across models, observing how granular data aligns or differs. It's like zooming in or out with a camera, where each zoom level represents a different level of detail for understanding causal effects.
Optimal Transport (OT) offers a mathematical framework for transforming one probability distribution into another in the most efficient way possible, often visualized as redistributing piles of sand. OT has had far-reaching implications, including in causality research, where it helps compare different causal models.
Causal Optimal Transport of Abstractions (COTA)
The paper proposes a new methodology called Causal Optimal Transport of Abstractions (COTA) that leverages observational and interventional data to learn about causal abstractions. Here are the key components of COTA:
- Multi-marginal formulation: COTA views the learning problem as a multi-marginal OT problem, where distributions from different interventions act as marginals in an optimization problem.
- Causal constraints: By incorporating causal knowledge through do-calculus constraints, the method ensures that the learned maps are not just statistically consistent but causally relevant.
- Algorithmic approach: The paper provides an algorithm for implementing COTA, which includes steps like computing the ω-cost matrix and running the COTA solver to obtain an optimal mapping.
Results and Implications
COTA was tested on synthetic and real-world data and outperformed traditional, non-causal methods, proving that it's not only theoretically sound but practically useful. The method demonstrated improved data augmentation in downstream tasks, suggesting its utility in a broad range of applications.
Additionally, the paper establishes a proof of the convexity of the presented optimization problem, ensuring that the obtained solution is indeed the best possible under the given constraints.
Impact of Causal Abstractions
The development of COTA represents a significant step in causal abstraction learning. It demonstrates how integrating causal knowledge into abstract mappings can help obtain better inferences and representations, even in the absence of complete SCM information. This approach may further enhance the fields of causality and AI, particularly in situations where causal relationships are represented at varying levels of detail.
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