Causal structure learning, also known as causal discovery, aims to estimate causal relationships between variables as a form of a causal directed acyclic graph (DAG) from observational data. One of the major frameworks is the order-based approach that first estimates a topological order of the underlying DAG and then prunes spurious edges from the fully-connected DAG induced by the estimated topological order. Previous studies often focus on the former ordering step because it can dramatically reduce the search space of DAGs. In practice, the latter pruning step is equally crucial for ensuring both computational efficiency and estimation accuracy. Most existing methods employ a pruning technique based on generalized additive models and hypothesis testing, commonly known as CAM-pruning. However, this approach can be a computational bottleneck as it requires repeatedly fitting additive models for all variables. Furthermore, it may harm estimation quality due to multiple testing. To address these issues, we introduce a new pruning method based on sparse additive models, which enables direct pruning of redundant edges without relying on hypothesis testing. We propose an efficient algorithm for learning sparse additive models by combining the randomized tree embedding technique with group-wise sparse regression. Experimental results on both synthetic and real datasets demonstrated that our method is significantly faster than existing pruning methods while maintaining comparable or superior accuracy.
In several scientific fields, such as biology and economics, it is often important to identify causal relationships between variables in observational data. Causal structure learning, also known as causal discovery, aims to estimate them as a form of a causal directed acyclic graph (DAG) (Pearl 2009). A causal DAG enables us to understand the relationships between variables and to predict the effect of interventions on the variables, which are crucial for decision-making in various applications (Peters, Janzing, and Schölkopf 2017).
One of the promising algorithmic frameworks for causal structure learning is the order-based approach (Teyssier and Koller 2005). Under some conditions on the data-generating process, previous studies have shown that the underlying causal DAG is identifiable from purely observational data (Peters et al. 2014). However, we need to search for a causal graph while ensuring it is acyclic, which incurs super-exponential computational costs in the number of variables (Chickering 1996). To alleviate this issue, the orderbased approach first estimates a topological order of the underlying causal DAG and then prunes spurious edges from the fully-connected DAG induced by the estimated topological order, as illustrated in Figure 1. By estimating a topological order in advance, we can obtain a causal DAG without explicitly imposing the acyclicity constraint, which reduces the search space dramatically (Rolland et al. 2022).
While existing studies often focus on the former ordering step, in practice, the latter pruning step is equally crucial for ensuring both computational efficiency and estimation accuracy. Most existing methods employ a pruning algorithm based on generalized additive models (GAMs) (Hastie and Tibshirani 1986), known as CAM-pruning (Bühlmann, Peters, and Ernest 2014). It first fits a GAM for regressing each variable on its candidate parents, and then identifies redundant parents by hypothesis testing for the fitted GAM (Marra and Wood 2011). However, the computational cost of fitting GAMs is generally expensive, especially for high-dimensional cases. Since CAM-pruning needs to repeatedly fit GAMs for all variables, it often becomes the bottleneck of the entire algorithm (Rolland et al. 2022;Montagna et al. 2023). Furthermore, CAM-pruning also requires repeating hypothesis testing, which can degrade the estimation accuracy due to multiple testing (Huang et al. 2018).
The goal of this paper is to propose an alternative pruning method that addresses the aforementioned limitations of existing pruning methods. It enables us to accelerate the existing order-based causal structure learning algorithms without compromising their estimation quality.
Our Contributions In this paper, we propose a new pruning method for order-based causal structure learning. Our key idea is to learn a sparse additive model (Ravikumar et al. 2009) that regresses each variable on its candidate parents, which enables us to directly prune redundant candidate parents without requiring hypothesis testing. To accelerate the process of fitting sparse additive models, we propose a new framework, named Sparse Additive Randomized TRee Ensemble (SARTRE), by combining the randomized tree embedding and group-wise sparse regression techniques. Our arXiv:2602.15306v1 [stat.ML] 17 Feb 2026
Observational data Fully-connected DAG
Pruned DAG topological order: X 2 , X 3 , X 1 , X 4 pruned edges: (X 2 , X 3 ), (X 3 , X 4 )
Step 1
Step 2
Figure 1: An overview of the order-based causal structure learning algorithm. Given an observational dataset, it first estimates a topological order of the underlying causal DAG. Then, it prunes spurious edges from the fully-connected DAG induced by the estimated topological order. This paper focuses on the latter step and aims to propose an efficient and accurate pruning method.
contributions are summarized as follows:
• We introduce a new efficient framework for learning a sparse additive model, named SARTRE. We consider a special case of the additive model, where the shape function for each variable is expressed as a linear combination of weighted indicator functions over a set of intervals.
We propose to generate a set of intervals by randomized tree embedding (Moosmann, Triggs, and Jurie 2006), and show that we can efficiently learn the sparse weight vector via group lasso regression (Yuan and Lin 2006). • We propose an efficient pruning method for order-based causal structure learning by leveraging our SARTRE framework. Given an estimated topological order, our method can efficiently prune redundant edges from the fully-connected DAG induced by the estimated topological order without requiring hypothesis testing. Our method can be combined with any causal ordering algorithm, such as SCORE (Rolland et al. 2022). • By numerical experiments on synthetic and real datasets, we demonstrated that our method achieved a significant speedup compared to existing pruning methods, incl
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