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Optimizing NOTEARS Objectives via Topological Swaps

This is an implementation of the following paper:

[1] C. Deng, K. Bello, B. Aragam, P. Ravikumar. (2023). Optimizing NOTEARS Objectives via Topological Swaps. ICML'23

If you find this code useful, please consider citing:

@inproceedings{deng2023optimizing,
  title={Optimizing NOTEARS Objectives via Topological Swaps},
  author={Chang, Deng and Kevin, Bello and Bryon, Aragam and Pradeep, Ravikumar},
  booktitle={Proceedings of the 40th International Conference on Machine Learning},
  year={2023}
}

Summary

In this paper, we propose a bi-level algorithm that solve a class of non-convex optimization problems that has emerged in the context of learning directed acyclic graphs (DAGs). These problems (e.g. NOTEARS) involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. Compared to the previous work, our algorithm: (1) Is guaranteed to find a local minimum or a KKT point under weaker conditions; (2) Achieves better score; (3) leads to siginificant improvement in structure recovery (e.g. SHD).

Requirements

  • Python 3.6+
  • numpy
  • scipy
  • python-igraph
  • torch: Only used for nonlinear
  • scikit-learn

Contents

  • Topo_linear.py - implementation of TOPO for linear models (also support Logistic losses).
  • Topo_nonlinear.py - implementation of TOPO for nonlinear models.
  • Topo_utils - implementation of support function for TOPO.
  • utils.py - graph simulation, data simulation, and accuracy evaluation.

Running TOPO

Use requirements.txt to install the dependencies (recommended to use virtualenv or conda). The simplest way to try out TOPO is to run a simple (linear) example:

$ git clone https://github.com/Duntrain/TOPO.git
$ cd TOPO/
$ pip3 install -r requirements.txt
$ python3 Topo_linear.py

or a simple (nonlinear) example. you can replace PRINT_ENABLED = True to PRINT_ENABLED = False in script to suppress the printing of intermediate results.

$ git clone https://github.com/Duntrain/TOPO.git
$ cd TOPO/
$ pip3 install -r requirements.txt
$ python3 Topo_nonlinear.py

The above runs the TOPO on a randomly generated 20-node Erdos-Renyi graph with 1000 samples. The output should look like the below:

{'fdr': 0.0, 'tpr': 1.0, 'fpr': 0.0, 'shd': 0, 'nnz': 20}

Extension

Users can take advantage of the framework of TOPO to test different structure equation model (SEM) by simply writing their own regression and score function (i.e. regress(X,y), score(X,W)) in Topo_linear.py. We have implemented those functions for Linear and Logistic model.

Acknowledgments

We thank the authors of the NOTEARS repo for making their code available. Part of our code is based on their implementation, specially the utils.py file and some code from their implementation of nonlinear models.