Local Optimal Adjustments Discovery (LOAD)

This is the official code repository for Local Causal Discovery for Statistically Efficient Causal Inference by Mátyás Schubert, Tom Claassen and Sara Magliacane.

Note

This branch contains a minimal and portable implementation of LOAD. Check out the aistats2026 branch to reproduce the results presented in the paper.

The LOAD algorithm is implemented in load.py and depends only on numpy, as listed in requirements.txt.

Examples

The following minimal examples show how to run load and interpret its results. Please also install causal-learn to run them.

In the following example, LOAD outputs that the causal effect is identifiable for treatment 0 and outcome 1 and can be estimated with optimal adjustment set {5,6}. LOAD returns no adjustment sets for the causal effect of 1 on 0 as it is identified as zero.

from causallearn.utils.cit import CIT
import numpy as np
import networkx as nx

from load import load

# Generate data
true_dag = np.array(
    [
        [0, 1, 0, 0, 1, 0, 0],
        [0, 0, 0, 0, 0, 0, 0],
        [1, 0, 0, 0, 1, 0, 0],
        [1, 0, 0, 0, 1, 0, 1],
        [0, 0, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
    ]
)
dummy_data = np.zeros_like(true_dag)
true_dag = nx.DiGraph(true_dag)
nx.draw_circular(true_dag, with_labels=True)

# Run LOAD
ci_test = CIT(data=dummy_data, method="d_separation", true_dag=true_dag)
result = load(data=dummy_data, alpha=0.01, ci_test=ci_test, targets=[1, 0])
print(result)

>> {'adj_sets': {(0, 1): [{5, 6}]}, 'identifiable': True}

---

In the following example, LOAD outputs that the causal effect is not identifiable. For treatment 4 and outcome 1, the locally valid parent adjustment sets are {2, 3} and {0, 2, 3}. LOAD returns no adjustment sets for the causal effect of 1 on 4, which means that it is identified as zero.

from causallearn.utils.cit import CIT
import numpy as np
import networkx as nx

from load import load

# Generate data
true_dag = np.array(
    [
        [0, 1, 0, 0, 1, 0, 0],
        [0, 0, 0, 0, 0, 0, 0],
        [1, 0, 0, 0, 1, 0, 0],
        [1, 0, 0, 0, 1, 0, 1],
        [0, 0, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
    ]
)
dummy_data = np.zeros_like(true_dag)
true_dag = nx.DiGraph(true_dag)
nx.draw_circular(true_dag, with_labels=True)

# Run LOAD
ci_test = CIT(data=dummy_data, method="d_separation", true_dag=true_dag)
result = load(data=dummy_data, alpha=0.01, ci_test=ci_test, targets=[1, 4])
print(result)

>> {'adj_sets': {(4, 1): [{2, 3}, {0, 2, 3}]}, 'identifiable': False}

---

In the following example, LOAD outputs that the causal effect is not identifiable, and both targets might be treatments. Thus, LOAD outputs the locally valid parent adjustment sets for both treatment 4 and outcome 0, and treatment 0 and outcome 4.

from causallearn.utils.cit import CIT
import numpy as np
import networkx as nx

from load import load

# Generate data
true_dag = np.array(
    [
        [0, 1, 0, 0, 1, 0, 0],
        [0, 0, 0, 0, 0, 0, 0],
        [1, 0, 0, 0, 1, 0, 0],
        [1, 0, 0, 0, 1, 0, 1],
        [0, 0, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
        [0, 1, 0, 0, 0, 0, 0],
    ]
)
dummy_data = np.zeros_like(true_dag)
true_dag = nx.DiGraph(true_dag)
nx.draw_circular(true_dag, with_labels=True)

# Run LOAD
ci_test = CIT(data=dummy_data, method="d_separation", true_dag=true_dag)
result = load(data=dummy_data, alpha=0.01, ci_test=ci_test, targets=[0, 4])
print(result)

>> {'adj_sets': {(0, 4): [{2, 3}], (4, 0): [{2, 3}]}, 'identifiable': False}

Citation

@article{schubert2025local,
  title={Local Causal Discovery for Statistically Efficient Causal Inference},
  author={Schubert, M{\'a}ty{\'a}s and Claassen, Tom and Magliacane, Sara},
  journal={arXiv preprint arXiv:2510.14582},
  year={2025}
}