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CURLS: Causal Rule Learning for Subgroups with Significant Treatment Effect

Jiehui Zhou, Linxiao Yang, Xingyu Liu, Xinyue Gu, Liang Sun, Wei Chen

TL;DR

CURLS addresses the challenge of identifying subgroups with significant causal effects by learning interpretable rules that describe these subgroups. It formulates causal rule learning as a discrete optimization problem over CNF-based antecedents, optimizing a log-profit objective f(R) that balances large treatment effects and small variance with rule-set interpretability. The key technical contribution is an iterative minorize-maximization algorithm that builds an approximate submodular lower bound and solves it via submodular optimization, enabling efficient discovery of high-impact subgroups. Experiments on synthetic and real data show CURLS achieving stronger estimated and true effects with lower variance, while producing concise, low-overlap rules that facilitate interpretation and practical decision-making.

Abstract

In causal inference, estimating heterogeneous treatment effects (HTE) is critical for identifying how different subgroups respond to interventions, with broad applications in fields such as precision medicine and personalized advertising. Although HTE estimation methods aim to improve accuracy, how to provide explicit subgroup descriptions remains unclear, hindering data interpretation and strategic intervention management. In this paper, we propose CURLS, a novel rule learning method leveraging HTE, which can effectively describe subgroups with significant treatment effects. Specifically, we frame causal rule learning as a discrete optimization problem, finely balancing treatment effect with variance and considering the rule interpretability. We design an iterative procedure based on the minorize-maximization algorithm and solve a submodular lower bound as an approximation for the original. Quantitative experiments and qualitative case studies verify that compared with state-of-the-art methods, CURLS can find subgroups where the estimated and true effects are 16.1% and 13.8% higher and the variance is 12.0% smaller, while maintaining similar or better estimation accuracy and rule interpretability. Code is available at https://osf.io/zwp2k/.

CURLS: Causal Rule Learning for Subgroups with Significant Treatment Effect

TL;DR

CURLS addresses the challenge of identifying subgroups with significant causal effects by learning interpretable rules that describe these subgroups. It formulates causal rule learning as a discrete optimization problem over CNF-based antecedents, optimizing a log-profit objective f(R) that balances large treatment effects and small variance with rule-set interpretability. The key technical contribution is an iterative minorize-maximization algorithm that builds an approximate submodular lower bound and solves it via submodular optimization, enabling efficient discovery of high-impact subgroups. Experiments on synthetic and real data show CURLS achieving stronger estimated and true effects with lower variance, while producing concise, low-overlap rules that facilitate interpretation and practical decision-making.

Abstract

In causal inference, estimating heterogeneous treatment effects (HTE) is critical for identifying how different subgroups respond to interventions, with broad applications in fields such as precision medicine and personalized advertising. Although HTE estimation methods aim to improve accuracy, how to provide explicit subgroup descriptions remains unclear, hindering data interpretation and strategic intervention management. In this paper, we propose CURLS, a novel rule learning method leveraging HTE, which can effectively describe subgroups with significant treatment effects. Specifically, we frame causal rule learning as a discrete optimization problem, finely balancing treatment effect with variance and considering the rule interpretability. We design an iterative procedure based on the minorize-maximization algorithm and solve a submodular lower bound as an approximation for the original. Quantitative experiments and qualitative case studies verify that compared with state-of-the-art methods, CURLS can find subgroups where the estimated and true effects are 16.1% and 13.8% higher and the variance is 12.0% smaller, while maintaining similar or better estimation accuracy and rule interpretability. Code is available at https://osf.io/zwp2k/.
Paper Structure (18 sections, 3 theorems, 13 equations, 3 figures, 6 tables, 3 algorithms)

This paper contains 18 sections, 3 theorems, 13 equations, 3 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Heterogeneous Treatment Effect Estimation
  4. Rule Learning
  5. Subgroup Discovery
  6. Preliminaries
  7. Problem Formulation
  8. Algorithm
  9. Causal Rule Set Learning
  10. MM Procedure
  11. Submodular Lower Bound Optimization
  12. Evaluation
  13. Quantitative Experiments
  14. Case Studies
  15. Discussion
  16. ...and 3 more sections

Key Result

Proposition 1

$\sigma_{\mathcal{R}}^2 \le \frac{\sum_{i \in \mathcal{D}_{\mathcal{R}}^+} w_i(Y_i - \mu_{(m)})^2}{\sum_{i \in \mathcal{D}_{\mathcal{R}}^+} w_i}$, where $\mu_{(m)}$ is the weighted mean of outcome of the previous step in the MM procedure.

Figures (3)

  • Figure 1: An illustrative toy example. There is only one covariate $X$, and the change in $Y$ can be informally thought of as the effect. The subgroup corresponding to Rule1 has a high effect and low variance, which the users expect to find.
  • Figure 2: Illustration of the proposed algorithm. (A) A causal rule set is learned from the observational data, and overlap penalities are applied to minimize the case where a unit is covered by multiple rules. (B) A single causal rule is solved by the MM framework, and the rule is improved by iteratively optimizing the surrogate lower bound of the original objective. (C) We prove an surrogate lower bound with submodular properties, allowing us to efficiently solve the surrogate optimization problem using efficient submodular optimization.
  • Figure 3: Scalability test. Training time scales linearly with the number of covariates.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3