.. note::
    :class: sphx-glr-download-link-note

    Click :ref:`here <sphx_glr_download_auto_examples_frank_wolfe_plot_vertex_overlap.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_frank_wolfe_plot_vertex_overlap.py:


Update Direction Overlap in Frank-Wolfe
========================================

This example quantifies how many times the Frank-Wolfe algorithm selects
the same extremal vertex (which will determine the update direction) twice
in a row. Selecting the same vertex twice in a row is symptomatic of a poor
step-size, as it implies that the last two updates could have been replaced
by a single update with larger step-size.



.. image:: /auto_examples/frank_wolfe/images/sphx_glr_plot_vertex_overlap_001.png
    :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    Running on the Gisette dataset
    Running backtracking variant
    Running DR step-size variant
    Running on the RCV1 dataset
    Running backtracking variant
    Running DR step-size variant
    Running on the Madelon dataset
    Running backtracking variant
    Running DR step-size variant
    Running on the Covtype dataset
    Running backtracking variant
    Running DR step-size variant





|


.. code-block:: default

    import copt as cp
    import matplotlib.pyplot as plt
    from matplotlib.ticker import MaxNLocator
    import numpy as np

    # datasets and their respective loading functions
    import copt.constraint
    import copt.loss

    datasets = [
        ("Gisette", cp.datasets.load_gisette),
        ("RCV1", cp.datasets.load_rcv1),
        ("Madelon", cp.datasets.load_madelon),
        ("Covtype", cp.datasets.load_covtype),
    ]


    fig, axes = plt.subplots(nrows=2, ncols=2, figsize=(10, 5))
    for ax, (dataset_title, load_data) in zip(axes.ravel(), datasets):
        print("Running on the %s dataset" % dataset_title)

        X, y = load_data()
        n_samples, n_features = X.shape

        l1_ball = copt.constraint.L1Ball(n_features / 2.0)
        f = copt.loss.LogLoss(X, y)
        x0 = np.zeros(n_features)

        for i, (step, label, marker) in enumerate(
            [["backtracking", "backtracking", "^"], ["DR", "DR step-size", "d"]]
        ):
            print("Running %s variant" % label)
            st_prev = []
            overlap = []

            def trace(kw):
                """Store vertex overlap during execution of the algorithm."""
                s_t = kw["update_direction"] + kw["x"]
                if st_prev:
                    # check if the vertex of this and the previous iterate
                    # coincide. Since these might be sparse vectors, we use
                    # sparse.linalg.norm to make the comparison
                    prev_overlap = overlap[-1]
                    if np.linalg.norm(st_prev[0] - s_t) == 0:
                        overlap.append(prev_overlap + 1)
                    else:
                        overlap.append(prev_overlap)
                    st_prev[0] = s_t
                else:
                    overlap.append(0)
                    st_prev.append(s_t)

            cp.minimize_frank_wolfe(
                f.f_grad,
                x0,
                l1_ball.lmo,
                callback=trace,
                max_iter=int(1e4),
                step=step,
                verbose=True,
                lipschitz=f.lipschitz,
            )
            ax.plot(overlap, label=label, marker=marker, markevery=7 + i)
            ax.yaxis.set_major_locator(MaxNLocator(integer=True))
            ax.legend()
        ax.set_xlabel("number of iterations")
        ax.set_ylabel("LMO overlap")
        ax.set_title(dataset_title)
        fig.tight_layout()  # otherwise the right y-label is slightly clipped
        ax.grid()
    # plt.legend()
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 197 minutes  42.808 seconds)

**Estimated memory usage:**  1911 MB


.. _sphx_glr_download_auto_examples_frank_wolfe_plot_vertex_overlap.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download

     :download:`Download Python source code: plot_vertex_overlap.py <plot_vertex_overlap.py>`



  .. container:: sphx-glr-download

     :download:`Download Jupyter notebook: plot_vertex_overlap.ipynb <plot_vertex_overlap.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_