This directory provides an implementation of the algorithms used to compute the hypergeometric tail pseudo-inverse, as well as the code used to produce all figures of the paper "Improving Generalization Bounds for VC Classes Using the Hypergeometric Tail Inversion" by Leboeuf, LeBlanc and Marchand (available on arXiv).

To run the scripts, one must first install the package and its requirements. To do so, run the following command from the root directory:

pip install .

Doing so will also provide you with the package hypergeo, which implements an algorithm to compute the hypergeometric tail pseudo-inverses.

The code was written to run on Python 3.8 or more recent version. The requirements are shown in the file requirements.txt and can be installed using the command:

pip install -r requirements.txt

The code is split into 2 parts: the 'hypergeo' package and the 'scripts' directory.

The hypergeo package implements the utilities regarding the hypergeometric distribution (to compute the tail and its inverse), the binomial distribution (reimplementing the inverse as the scipy version suffered from numerical unstabilities) and some generalization bounds.

The scripts files produce the figures found in the paper using the hypergeo package. All figures are generated directly in LaTeX using the package python2latex. To run a script, navigate from the command line to the directory root directory of the project and run the command

python "./scripts/<dir_name>/<file_name>.py"

The code does not provide command line control on the parameters of each script. However, each script is fairly simple, and parameters can be directly changed in the __main__ part of the script.

• Section 3.3: The ghost sample trade-off. In this section, we claim that optimizing m' gives relative gain between 8% and 10%. To obtain these number, you need to run the file mprime_tradeoff/generate_mprime_data.py to first generate the data, and then run mprime_tradeoff/stats.py.

• Section 5: Numerical comparison. Figure 1a and 1b are obtain by executing the scripts bounds_comparison/bounds_comparison_risk.py and bounds_comparison/bounds_comparison_d.py respectively. Figure 2a and 2b are obtain by executing the scripts bounds_comparison/bounds_comparison_m.py, the first setting the variable risk to 0, the second by setting it equal to 0.1.

• Appendix B: Overview of the hypergeometric distribution. Figure 3 is generated from hypergeometric_tail/hyp_tail_plot.py. Figure 4 is generated from hypergeometric_tail/hyp_tail_inv_plot.py. Algorithm 1 is implemented in the hypergeo file hypergeo/hypergeometric_distribution.py as the function hypergeometric_tail_inverse. Algorithm 2 is implemented in the hypergeo file hypergeo/hypergeometric_distribution.py as the function berkopec_hypergeometric_tail_inverse.

• Appendix D: In-depth analysis of the ghost sample trade-off. Figure 5 is generated from mprime_tradeoff/plot_epsilon_comp.py. Figure 6 is generated from mprime_tradeoff/plot_mprime_best.py.

• Appendix E: The hypergeometric tail inversion relative deviation bound. To generate Figure 7 and 8, you must first run the file relative_deviation_mprime_tradeoff/mprime_tradeoff_relative_deviation.py to generate the data, then run the script relative_deviation_mprime_tradeoff/plot_epsilon_comp.py to produce Figure 7 and relative_deviation_comparison/plot_mprime_best.py to produce Figure 8.

• Appendix G: The hypergeometric tail lower bound . Figure 9 is generated from lower_bound/lower_bound_comparison_risk.py.

• Appendix F: Further numerical comparisons. Figure 10 and 12a are generated from bounds_comparison/bounds_comparison_risk.py by changing the parameters of the scripts. Figure 11 and 12b is generated from bounds_comparison/bounds_comparison_m.py by changing the parameters of the scripts. Figure 13a and 13b are generated from bounds_comparison/sample_compression_comparison_risk.py and bounds_comparison/sample_compression_comparison_m.py respectively.

The script pseudo-inverse_benchmarking/pseudo-inverse_benchmarking.py benchmarks the various algorithms used to invert the hypergeometric tail. The 'tests' directory contains unit tests using the package pytest.