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Causal Unit Selection using Tractable Arithmetic Circuits

Haiying Huang, Adnan Darwiche

TL;DR

This work tackles unit selection under causal objectives by reducing the optimization to Reverse-MAP (R-MAP) on an objective model built from triplet copies of the SCM and a mixture node, yielding a single observational probability representation. It then compiles the objective model into tractable arithmetic circuits (ACs) and leverages a two-pass, division-based R-MAP on ACs to achieve linear-time optimization with respect to circuit size. Empirically, ACE_RMAP substantially outperforms VE_RMAP on random, dense SCMs, enabling exact unit selection for larger problems and higher constrained treewidth than previously feasible. The approach significantly advances scalable causal reasoning for selecting optimal units, policies, or interventions in complex settings where traditional inference-based methods falter.

Abstract

The unit selection problem aims to find objects, called units, that optimize a causal objective function which describes the objects' behavior in a causal context (e.g., selecting customers who are about to churn but would most likely change their mind if encouraged). While early studies focused mainly on bounding a specific class of counterfactual objective functions using data, more recent work allows one to find optimal units exactly by reducing the causal objective to a classical objective on a meta-model, and then applying a variant of the classical Variable Elimination (VE) algorithm to the meta-model -- assuming a fully specified causal model is available. In practice, however, finding optimal units using this approach can be very expensive because the used VE algorithm must be exponential in the constrained treewidth of the meta-model, which is larger and denser than the original model. We address this computational challenge by introducing a new approach for unit selection that is not necessarily limited by the constrained treewidth. This is done through compiling the meta-model into a special class of tractable arithmetic circuits that allows the computation of optimal units in time linear in the circuit size. We finally present empirical results on random causal models that show order-of-magnitude speedups based on the proposed method for solving unit selection.

Causal Unit Selection using Tractable Arithmetic Circuits

TL;DR

This work tackles unit selection under causal objectives by reducing the optimization to Reverse-MAP (R-MAP) on an objective model built from triplet copies of the SCM and a mixture node, yielding a single observational probability representation. It then compiles the objective model into tractable arithmetic circuits (ACs) and leverages a two-pass, division-based R-MAP on ACs to achieve linear-time optimization with respect to circuit size. Empirically, ACE_RMAP substantially outperforms VE_RMAP on random, dense SCMs, enabling exact unit selection for larger problems and higher constrained treewidth than previously feasible. The approach significantly advances scalable causal reasoning for selecting optimal units, policies, or interventions in complex settings where traditional inference-based methods falter.

Abstract

The unit selection problem aims to find objects, called units, that optimize a causal objective function which describes the objects' behavior in a causal context (e.g., selecting customers who are about to churn but would most likely change their mind if encouraged). While early studies focused mainly on bounding a specific class of counterfactual objective functions using data, more recent work allows one to find optimal units exactly by reducing the causal objective to a classical objective on a meta-model, and then applying a variant of the classical Variable Elimination (VE) algorithm to the meta-model -- assuming a fully specified causal model is available. In practice, however, finding optimal units using this approach can be very expensive because the used VE algorithm must be exponential in the constrained treewidth of the meta-model, which is larger and denser than the original model. We address this computational challenge by introducing a new approach for unit selection that is not necessarily limited by the constrained treewidth. This is done through compiling the meta-model into a special class of tractable arithmetic circuits that allows the computation of optimal units in time linear in the circuit size. We finally present empirical results on random causal models that show order-of-magnitude speedups based on the proposed method for solving unit selection.
Paper Structure (14 sections, 4 theorems, 3 equations, 2 figures, 1 table)

This paper contains 14 sections, 4 theorems, 3 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unit Selection by Reduction to Reverse-MAP
  3. Objective Model
  4. Reverse-MAP (R-MAP)
  5. Basics of Arithmetic Circuits
  6. Solving MAP using Arithmetic Circuits
  7. Circuits for Reverse-MAP
  8. Circuit Division
  9. R-MAP
  10. Empirical Results
  11. Problem Generation
  12. Results
  13. Conclusion
  14. Acknowledgement

Key Result

Proposition 1

Consider a decision-AC over variables ${\bf X}$ and let ${\bf U} \subseteq {\bf X}$. If the circuit satisfies: 1) no $+$-node $n$ with $\mathop{\mathrm{dvar}}\nolimits(n) \in {\bf U}$ is below some $+$-node $m$ with $\mathop{\mathrm{dvar}}\nolimits(m) \notin {\bf U}$ and 2) every indicator $\lambda_

Figures (2)

  • Figure 1: Example showing the objective model for a causal objective function with 2 components (Huang and Darwiche 2023).
  • Figure 2: An AC that computes factor $f(A,B)$.

Theorems & Definitions (8)

  • Definition 1: Decision-AC
  • Proposition 1
  • Definition 2
  • Proposition 2
  • proof
  • Theorem 1
  • Definition 3
  • Theorem 2