Multiclass Neural Network Minimization via Tropical Newton Polytope Approximation

Georgios Smyrnis, Petros Maragos

This is the repository for the code of our ICML 2020 paper, "Multiclass Neural Network Minimization via Tropical Newton Polytope Approximation". The provided code produces the results for the experiments, as defined in the paper. Moreover, a script is also provided, which creates the figures presented in the paper.

Citation

If you use this code, please cite the following:

@InProceedings{SmyrnisMaragos_2020_ICML,
author = {Smyrnis, Georgios and Maragos, Petros},
title = {Multiclass Neural Network Minimization via Tropical Newton Polytope Approximation},
booktitle = {Proceedings of the 37th International Conference on Machine Learning (ICML)},
publisher = {PMLR},
month = {July},
year = {2020}
} 

Datasets Used

For our experiments we used the following datasets. We encourage anyone interested to visit the original sources and cite the appropriate references if they make use of these datasets.

• MNIST
• Fashion-MNIST

File Description:

The following files are contained:

• mnist_training.py: Trains the base model to be minimized, for the MNIST dataset.
• fashion_mnist_training.py: Trains the base model to be minimized, for the Fashion-MNIST dataset.
• multiclass_minim_one_vs_all_heuristic.py: Performs the One-Vs-All multiclass minimization procedure, outlined in Section 4.2, based on the single output minimizaton algorithm from Section 3.2.
• multiclass_minim_one_vs_all_stable.py: Performs the same minimization procedure, but using as single output minimization algorithm the one from Section 5.1.
• result_aggregation_mnist.py: Aggregates results from the experimental runs, calculating mean and standard deviation (note that the same code is used for both datasets, see guidelines below).
• create_figures.py: Creates the figures presented in the paper.
• run_test_one_vs_all_mnist.sh: Script which runs the experiments for MNIST.
• run_test_one_vs_all_fashion.sh: Script which runs the experiments for Fashion-MNIST.

Experimental Section:

For the MNIST experiments (Tables 1, 2), run the following three commands:

• ./run_test_one_vs_all.sh
• python result_aggregation_mnist.py ./results_one_vs_all_mnist simple
• python result_aggregation_mnist.py ./results_one_vs_all_mnist extra

For the Fashion-MNIST experiments (Table 3), run the following two commands:

• ./run_test_one_vs_all_fashion.sh
• python result_aggregation_mnist.py ./results_one_vs_all_fashion extra

The results can be found as follows ("simple" corresponds to the method of Section 4.2 with the Heuristic Minimization, and "extra" to the use of the Stable Minimization of Section 5.1, as explained in the experimental section of the paper):

• Table 1: ./results_one_vs_all_mnist/results_simple/
• Table 2: ./results_one_vs_all_mnist/results_extra/
• Table 3: ./results_one_vs_all_fashion/results_extra/

Output format:

For the files without the _std mark:

• Column 1: Percentages.
• Column 2: Average Score.

For the files with the _std mark:

• Column 2: Standard deviation (for the percentage corresponding to each row).

Note that the folders named results_<simple/extra>_<N> contain results from the individual trials.

Figures:

For the figures, run the following command (each figure will be presented and can then be saved manually, in the desired format):

python create_figures.py

Further Details - Requirements:

These experiments were run using Python 3.7.6, on an Ubuntu 18.04 OS. The following packages were installed on the testing environment and used in the provided code:

• numpy: v1.18.1
• scipy: v1.3.2
• tensorflow: v1.15.0
• keras: v2.2.4
• tqdm: v4.41.1
• pandas: v0.25.3
• matplotlib: v3.1.3

All of the dependencies of the above were also installed in the testing environment, as well as scikit-learn, v0.22.1. Installing these packages, along with their respective dependencies, leads to an environment equivalent to the one on which the experiments were run. Note that this is not a strict requirement; it might be possible to run the above experiments in a different configuration.

References

[1] G.Smyrnis and P. Maragos. Multiclass Neural Network Minimization via Tropical Newton Polytope Approximation. ICML 2020.

[2] F. Chollet et al. Keras. 2015.

[3] Y. Lecun, L. Bottou, Y. Bengio and P. Haffner. Gradient-Based Learning Applied to Document Recognition. Proc. of the IEEE 1998.

[4] H. Xiao, K. Rasul and R. Vollgraf. Fashion-MNIST: A Novel Image Dataset for Benchmarking Machine Learning Algorithms. arXiv 2017.

License

Our code is released under the MIT license.