guillaumew16 / particle-MNE

Code for "An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games"

Code for "An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games"

Set up

Requires julia and jupyter notebook. The required julia packages are listed in the project file Project.toml.

Detailed steps for a possible setup using conda

1. Install conda and julia. (Tested with conda v4.14.0, julia v1.6.3.)
2. In Unix shell:

conda create -n particle_MNE
conda activate particle_MNE
conda install jupyter # or conda install -c conda-forge jupyterlab

3. In julia REPL:

using Pkg # or type ] to enter Pkg REPL-mode
pkg"activate ."
pkg"instantiate"

4. In Unix shell:

# conda activate particle_MNE
jupyter notebook # or jupyter lab

Reproducing the figures of the paper

Run the notebooks in the directory generate_plots/. The figures are saved in png format to the relevant subdirectory of results/.

The notebooks are almost identical to the ones described below (e.g., generate_plots/particle_random_fourier_1D_genplots.ipynb is almost identical to particle_random_fourier_1D.ipynb), with adjustments only to the plotting parameters.

Experiments

Set extrasteps=1 for CP-MDA (or Mirror Descent-Ascent for finite games), extrasteps=2 for CP-MP (or Mirror Prox for finite games).

The dependency of the exponential convergence rate with respect to the step-size can be tested easily by setting scaling and varying alpha, and checking whether the slope of the NI error is independent of alpha; equivalently (provided that the local convergence regime always kicks in immediately), whether the NI error at the last iterate is independent of alpha.

Finite games (Mirror Descent-Ascent, Mirror Prox)

• Notebook weightonly_gaussian_gmat.ipynb: finite game with random payoff matrix with i.i.d Gaussian components.
• Notebook weightonly_random_fourier_1D.ipynb: finite game with random payoff matrix obtained by discretizing a periodic Gaussian process payoff function, in dimension dimx=1 and dimy=1, on a uniform grid of $[0,1] \times [0,1]$.

Make use of functions defined in utils/ and in weightonly/.

Payoff functions drawn from a periodic Gaussian process (dimx=dimy=1)

Notebook particle_random_fourier_1D.ipynb. Makes use of functions defined in utils/ and in particle_1D/.

A synthetic example where the continuous-time flow of CP-PP does not converge

Notebook particle_fourier_1D_counterexample.ipynb. Makes use of functions defined in utils/ and in particle_1D/.

Max-F1-margin classification with 2LNNs

Notebook maxF1margin.ipynb. The code is self-contained.

Distributionally-robust classification with 2LNNs

Notebook distrib_robust.ipynb. The code is self-contained.

Additional code

For reference we also provide an implementation of CP-MP for computing the MNE of continuous games with payoff functions drawn from a periodic Gaussian process in dimension dimx, dimy > 1 in the notebook particle_random_fourier_multidim, which uses functions defined in particle_multidim/. It is possible to check manually that the CP-MP iterates converge to a sparse measure, but evaluating their NI errors accurately takes a very long time.