Dags With No Tears Continuous Optimization for Structure Learning

DAGs with NO TEARS 🚫 💧

This is an implementation of the following papers:

[1] Zheng, X., Aragam, B., Ravikumar, P., & Xing, E. P. (2018). DAGs with NO TEARS: Continuous optimization for structure learning (NeurIPS 2018, Spotlight).

[2] Zheng, X., Dan, C., Aragam, B., Ravikumar, P., & Xing, E. P. (2020). Learning sparse nonparametric DAGs (AISTATS 2020, to appear).

If you find this code useful, please consider citing:

              @inproceedings{zheng2018dags,     author = {Zheng, Xun and Aragam, Bryon and Ravikumar, Pradeep and Xing, Eric P.},     booktitle = {Advances in Neural Information Processing Systems},     title = {{DAGs with NO TEARS: Continuous Optimization for Structure Learning}},     year = {2018} }

              @inproceedings{zheng2020learning,     author = {Zheng, Xun and Dan, Chen and Aragam, Bryon and Ravikumar, Pradeep and Xing, Eric P.},     booktitle = {International Conference on Artificial Intelligence and Statistics},     title = {{Learning sparse nonparametric DAGs}},     year = {2020} }

Update

Code for nonlinear NOTEARS has been added. See [2] for details.

tl;dr Structure learning in <60 lines

Check out linear.py for a complete, end-to-end implementation of the NOTEARS algorithm in fewer than 60 lines.

This includes L2, Logistic, and Poisson loss functions with L1 penalty.

Introduction

A directed acyclic graphical model (aka Bayesian network) with d nodes defines a distribution of random vector of size d. We are interested in the Bayesian Network Structure Learning (BNSL) problem: given n samples from such distribution, how to estimate the graph G?

A major challenge of BNSL is enforcing the directed acyclic graph (DAG) constraint, which is combinatorial. While existing approaches rely on local heuristics, we introduce a fundamentally different strategy: we formulate it as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. In other words,

characterization

where h is a smooth function whose level set exactly characterizes the space of DAGs.

Requirements

Python 3.6+
numpy
scipy
python-igraph: Install igraph C core and pkg-config first.
torch: Optional, only used for nonlinear model.

Contents (New version)

linear.py - the 60-line implementation of NOTEARS with l1 regularization for various losses
nonlinear.py - nonlinear NOTEARS using MLP or basis expansion
locally_connected.py - special layer structure used for MLP
lbfgsb_scipy.py - wrapper for scipy's LBFGS-B
utils.py - graph simulation, data simulation, and accuracy evaluation

Running a simple demo

The simplest way to try out NOTEARS is to run a simple example:

$ git clone https://github.com/xunzheng/notears.git $              cd              notears/ $ python notears/linear.py

This runs the l1-regularized NOTEARS on a randomly generated 20-node Erdos-Renyi graph with 100 samples. Within a few seconds, you should see output like this:

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

The data, ground truth graph, and the estimate will be stored in X.csv, W_true.csv, and W_est.csv.

Running as a command

Alternatively, if you have a CSV data file X.csv, you can install the package and run the algorithm as a command:

$ pip install git+git://github.com/xunzheng/notears $ notears_linear X.csv

The output graph will be stored in W_est.csv.

Examples: Erdos-Renyi graph

Ground truth: d = 20 nodes, 2d = 40 expected edges.
Estimate with n = 1000 samples: lambda = 0, lambda = 0.1, and FGS (baseline).

Both lambda = 0 and lambda = 0.1 are close to the ground truth graph when n is large.
Estimate with n = 20 samples: lambda = 0, lambda = 0.1, and FGS (baseline).

When n is small, lambda = 0 perform worse while lambda = 0.1 remains accurate, showing the advantage of L1-regularization.

Examples: Scale-free graph

Ground truth: d = 20 nodes, 4d = 80 expected edges.

The degree distribution is significantly different from the Erdos-Renyi graph. One nice property of our method is that it is agnostic about the graph structure.
Estimate with n = 1000 samples: lambda = 0, lambda = 0.1, and FGS (baseline).

The observation is similar to Erdos-Renyi graph: both lambda = 0 and lambda = 0.1 accurately estimates the ground truth when n is large.
Estimate with n = 20 samples: lambda = 0, lambda = 0.1, and FGS (baseline).

Similarly, lambda = 0 suffers from small n while lambda = 0.1 remains accurate, showing the advantage of L1-regularization.

Other implementations

Python: https://github.com/jmoss20/notears
Tensorflow with Python: https://github.com/ignavier/notears-tensorflow

Dags With No Tears Continuous Optimization for Structure Learning