.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/svm/plot_svm_scale_c.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via JupyterLite or Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_svm_plot_svm_scale_c.py: ============================================== Scaling the regularization parameter for SVCs ============================================== The following example illustrates the effect of scaling the regularization parameter when using :ref:`svm` for :ref:`classification `. For SVC classification, we are interested in a risk minimization for the equation: .. math:: C \sum_{i=1, n} \mathcal{L} (f(x_i), y_i) + \Omega (w) where - :math:`C` is used to set the amount of regularization - :math:`\mathcal{L}` is a `loss` function of our samples and our model parameters. - :math:`\Omega` is a `penalty` function of our model parameters If we consider the loss function to be the individual error per sample, then the data-fit term, or the sum of the error for each sample, increases as we add more samples. The penalization term, however, does not increase. When using, for example, :ref:`cross validation `, to set the amount of regularization with `C`, there would be a different amount of samples between the main problem and the smaller problems within the folds of the cross validation. Since the loss function dependens on the amount of samples, the latter influences the selected value of `C`. The question that arises is "How do we optimally adjust C to account for the different amount of training samples?" .. GENERATED FROM PYTHON SOURCE LINES 36-41 .. code-block:: Python # Author: Andreas Mueller # Jaques Grobler # License: BSD 3 clause .. GENERATED FROM PYTHON SOURCE LINES 42-51 Data generation --------------- In this example we investigate the effect of reparametrizing the regularization parameter `C` to account for the number of samples when using either L1 or L2 penalty. For such purpose we create a synthetic dataset with a large number of features, out of which only a few are informative. We therefore expect the regularization to shrink the coefficients towards zero (L2 penalty) or exactly zero (L1 penalty). .. GENERATED FROM PYTHON SOURCE LINES 51-59 .. code-block:: Python from sklearn.datasets import make_classification n_samples, n_features = 100, 300 X, y = make_classification( n_samples=n_samples, n_features=n_features, n_informative=5, random_state=1 ) .. GENERATED FROM PYTHON SOURCE LINES 60-70 L1-penalty case --------------- In the L1 case, theory says that provided a strong regularization, the estimator cannot predict as well as a model knowing the true distribution (even in the limit where the sample size grows to infinity) as it may set some weights of otherwise predictive features to zero, which induces a bias. It does say, however, that it is possible to find the right set of non-zero parameters as well as their signs by tuning `C`. We define a linear SVC with the L1 penalty. .. GENERATED FROM PYTHON SOURCE LINES 70-75 .. code-block:: Python from sklearn.svm import LinearSVC model_l1 = LinearSVC(penalty="l1", loss="squared_hinge", dual=False, tol=1e-3) .. GENERATED FROM PYTHON SOURCE LINES 76-78 We compute the mean test score for different values of `C` via cross-validation. .. GENERATED FROM PYTHON SOURCE LINES 78-108 .. code-block:: Python import numpy as np import pandas as pd from sklearn.model_selection import ShuffleSplit, validation_curve Cs = np.logspace(-2.3, -1.3, 10) train_sizes = np.linspace(0.3, 0.7, 3) labels = [f"fraction: {train_size}" for train_size in train_sizes] shuffle_params = { "test_size": 0.3, "n_splits": 150, "random_state": 1, } results = {"C": Cs} for label, train_size in zip(labels, train_sizes): cv = ShuffleSplit(train_size=train_size, **shuffle_params) train_scores, test_scores = validation_curve( model_l1, X, y, param_name="C", param_range=Cs, cv=cv, n_jobs=2, ) results[label] = test_scores.mean(axis=1) results = pd.DataFrame(results) .. GENERATED FROM PYTHON SOURCE LINES 109-136 .. code-block:: Python import matplotlib.pyplot as plt fig, axes = plt.subplots(nrows=1, ncols=2, sharey=True, figsize=(12, 6)) # plot results without scaling C results.plot(x="C", ax=axes[0], logx=True) axes[0].set_ylabel("CV score") axes[0].set_title("No scaling") for label in labels: best_C = results.loc[results[label].idxmax(), "C"] axes[0].axvline(x=best_C, linestyle="--", color="grey", alpha=0.7) # plot results by scaling C for train_size_idx, label in enumerate(labels): train_size = train_sizes[train_size_idx] results_scaled = results[[label]].assign( C_scaled=Cs * float(n_samples * np.sqrt(train_size)) ) results_scaled.plot(x="C_scaled", ax=axes[1], logx=True, label=label) best_C_scaled = results_scaled["C_scaled"].loc[results[label].idxmax()] axes[1].axvline(x=best_C_scaled, linestyle="--", color="grey", alpha=0.7) axes[1].set_title("Scaling C by sqrt(1 / n_samples)") _ = fig.suptitle("Effect of scaling C with L1 penalty") .. image-sg:: /auto_examples/svm/images/sphx_glr_plot_svm_scale_c_001.png :alt: Effect of scaling C with L1 penalty, No scaling, Scaling C by sqrt(1 / n_samples) :srcset: /auto_examples/svm/images/sphx_glr_plot_svm_scale_c_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 137-155 In the region of small `C` (strong regularization) all the coefficients learned by the models are zero, leading to severe underfitting. Indeed, the accuracy in this region is at the chance level. Using the default scale results in a somewhat stable optimal value of `C`, whereas the transition out of the underfitting region depends on the number of training samples. The reparametrization leads to even more stable results. See e.g. theorem 3 of :arxiv:`On the prediction performance of the Lasso <1402.1700>` or :arxiv:`Simultaneous analysis of Lasso and Dantzig selector <0801.1095>` where the regularization parameter is always assumed to be proportional to 1 / sqrt(n_samples). L2-penalty case --------------- We can do a similar experiment with the L2 penalty. In this case, the theory says that in order to achieve prediction consistency, the penalty parameter should be kept constant as the number of samples grow. .. GENERATED FROM PYTHON SOURCE LINES 155-175 .. code-block:: Python model_l2 = LinearSVC(penalty="l2", loss="squared_hinge", dual=True) Cs = np.logspace(-8, 4, 11) labels = [f"fraction: {train_size}" for train_size in train_sizes] results = {"C": Cs} for label, train_size in zip(labels, train_sizes): cv = ShuffleSplit(train_size=train_size, **shuffle_params) train_scores, test_scores = validation_curve( model_l2, X, y, param_name="C", param_range=Cs, cv=cv, n_jobs=2, ) results[label] = test_scores.mean(axis=1) results = pd.DataFrame(results) .. GENERATED FROM PYTHON SOURCE LINES 176-202 .. code-block:: Python import matplotlib.pyplot as plt fig, axes = plt.subplots(nrows=1, ncols=2, sharey=True, figsize=(12, 6)) # plot results without scaling C results.plot(x="C", ax=axes[0], logx=True) axes[0].set_ylabel("CV score") axes[0].set_title("No scaling") for label in labels: best_C = results.loc[results[label].idxmax(), "C"] axes[0].axvline(x=best_C, linestyle="--", color="grey", alpha=0.8) # plot results by scaling C for train_size_idx, label in enumerate(labels): results_scaled = results[[label]].assign( C_scaled=Cs * float(n_samples * np.sqrt(train_sizes[train_size_idx])) ) results_scaled.plot(x="C_scaled", ax=axes[1], logx=True, label=label) best_C_scaled = results_scaled["C_scaled"].loc[results[label].idxmax()] axes[1].axvline(x=best_C_scaled, linestyle="--", color="grey", alpha=0.8) axes[1].set_title("Scaling C by sqrt(1 / n_samples)") fig.suptitle("Effect of scaling C with L2 penalty") plt.show() .. image-sg:: /auto_examples/svm/images/sphx_glr_plot_svm_scale_c_002.png :alt: Effect of scaling C with L2 penalty, No scaling, Scaling C by sqrt(1 / n_samples) :srcset: /auto_examples/svm/images/sphx_glr_plot_svm_scale_c_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 203-211 For the L2 penalty case, the reparametrization seems to have a smaller impact on the stability of the optimal value for the regularization. The transition out of the overfitting region occurs in a more spread range and the accuracy does not seem to be degraded up to chance level. Try increasing the value to `n_splits=1_000` for better results in the L2 case, which is not shown here due to the limitations on the documentation builder. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 17.262 seconds) .. _sphx_glr_download_auto_examples_svm_plot_svm_scale_c.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/svm/plot_svm_scale_c.ipynb :alt: Launch binder :width: 150 px .. container:: lite-badge .. image:: images/jupyterlite_badge_logo.svg :target: ../../lite/lab/?path=auto_examples/svm/plot_svm_scale_c.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_svm_scale_c.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_svm_scale_c.py ` .. include:: plot_svm_scale_c.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_