#LinearFW

This is the code to reproduce all the experiments in our NIPS 2015 paper:

On the Global Linear Convergence of Frank-Wolfe Optimization Variants
Simon Lacoste-Julien and Martin Jaggi
NIPS 2015

which covers the global linear convergence rate of Frank-Wolfe optimization variants for problems described as in Eq. (1) in the paper. It contains the implementation of Frank-Wolfe, away-steps Frank-Wolfe and pairwise Frank-Wolfe on two applications:

1. l1-constrained least-square regression (lasso);
2. a QP on the flow polytope coming from a video co-localization application.

The code runs in Matlab (was tested in Matlab 2014 on Linux, Windows and Mac). But for the first two folders below, it also runs in Octave easily by removing the line initializing the random seed.

There are three folders:

• FW_lasso contains the Lasso experiment to produce the top figure in Figure 2. Launch run_FW.m in the folder to produce the plot (takes a few seconds).
• triangle_FW_experiment contains the empirical tigthness of the linear rate constant experiment, from Appendix E (Figure 5). Launch run_triangles.m in the folder to produce the plots (takes about 30 seconds).
• FW_video_colocalization containts the video co-localization QP experiment to produce the bottom figure in Figure 2. Launch run_FW.m in the folder to produce the plot (takes less than 1 minute). If you get the Undefined function 'solver_video_mex' error, you need to mex the following file to get the correct LMO; go to the solvers subfolder, and then run mex solver_video_mex.cpp in it.

##Credits

The video co-localization code was written by Armand Joulin and Kevin Tang. We obtained it here and modified the Frank-Wolfe code by adding a hashing function to make the active set maintenance more efficient, as well as added the pairwise FW variant. Their video co-localization approach is described in the paper:

Efficient Image and Video Co-localization with Frank-Wolfe Algorithm
Armand Joulin, Kevin Tang and Li Fei-Fei
ECCV 2014

##Extending the code

You can easily re-use the code in the FW_video_colocalization folder to adapt it to other QPs with different domains. For this, you mainly need to modify two things:

1. You need to implement your own Linear Minimization Oracle (LMO) on your domain. You pass it as the fun_optim argument to the FW functions (FW for standard FW; AFW for away-steps FW and PFW for pairwise FW). This function takes a vector as argument and returns an atom that minimizes the dot product with this vector over the domain.
2. If your atoms are something different than just 0-1 vectors, then you also need to modify the hashing internal function for AFW and PFW so that you can properly encode the atoms in your domain in a unique string (or number) (this is used for the efficient maintenance of the active set). The current hashing function only supports 0-1 vectors.

##Disclaimer

This code is not meant to be the most efficient. For example, an implementation handling the sparsity of the atoms could be faster for the video co-localization application. Our goal was to simply demonstrate how the FW variants work on practical applications.

##Citation

Please use the following BibTeX entry to cite this software in your work:

@inproceedings{LacosteJulien2015linearFW,
  author    = {Simon Lacoste-Julien and Martin Jaggi},
  title     = {On the Global Linear Convergence of {F}rank-{W}olfe Optimization Variants},
  booktitle = {Advances in Neural Information Processing Systems (NIPS)},
  year      = {2015},
}

And if you use the video co-localization LMO, you also need to cite:

@inproceedings{JouTangFeiECCV14,
  title     = {Efficient Image and Video Co-localization with {F}rank-{W}olfe Algorithm},
  author    = {Armand Joulin and Kevin Tang and Li Fei-Fei},
  booktitle = {European Conference on Computer Vision (ECCV)},
  year      = {2014},
}

##Authors

• Simon Lacoste-Julien
• Martin Jaggi