Supplementary Material of "Constraint-Free Structure Learning with Smooth Acyclic Orientations" @ICLR 2024

Environment

The conda environment is specified in envs/environment.yml. Therefore, to install the environment, run the following command:

conda env create -f envs/environment.yml

Synthetic Dataset Notation

We use the notation n{samples}_d{nodes}_{ER\|SF}{edge_factor}_{noise} to denote syntethic datasets. Therefore, the dataset n1000_d100_SF6_gauss has 1000 samples, on a scale-free graph with 100 nodes and edge factor 6, and the noise is gaussian. We admit the following noise terms: gauss, exp, and gumbel for the linear case and mlp for the non-linear case.

Experimental run

For each experiment, run the following command:

python -m benchmark grid {model} {dataset} config/grids/{grid_name} --n_grid_samples={NSAMPLES} --num_cpus={NCPUS} --n_repetitions=5

The command performs a grid search, whose results are store in the grids folder, and validates the best configuration (according to the ROCAUC metric), whose results are stored in the validation folder. Both files are named {dataset}_{grid_name}.csv. We include the results of our runs in the grids and validation directories.

The available model names are:

• cosmo
• dagma
• nocurl
• notears

The hyperparameter ranges for the grid search in the config/grids/ directory are:

• cosmo.yaml; Linear COSMO
• cosmo_nl.yaml; Non-Linear COSMO
• dagma.yaml; Linear DAGMA
• dagma_nl.yaml; Non-Linear DAGMA
• nocurl.yaml; Linear NOCURL
• nocurl_joint.yaml; Linear NOCURL-U, i.e., the unconstrained variant.
• notears.yaml; Linear NOTEARS

For instance, to run non-linear cosmo on n1000_d20_ER4_mlp, the command would be:

python -m benchmark grid cosmo n1000_d20_ER4_mlp config/grids/cosmo_nl.yaml --n_grid_samples=200 --num_cpus=50 --n_repetitions=5

The table.ipynb notebook can be used to generate the tables in the paper by selecting the number of nodes, the edge factor and the noise type.

Step time

The step time simulation experiment can be replicated for each model by running the following command:

python steptime.py cosmo

The model names are, as before, cosmo, dagma, nocurl, and notears.

Each run produces a {modelname}.csv file in the steptime directory. Then, the steptime.ipynb notebook can be used to concatenate and plot the results across all models.