1.4. Support Vector Machines¶

1.4.1.1. Multi-class classification¶

SVC and NuSVC implement the “one-versus-one” approach for multi-class classification. In total, n_classes * (n_classes - 1) / 2 classifiers are constructed and each one trains data from two classes. To provide a consistent interface with other classifiers, the decision_function_shape option allows to monotonically transform the results of the “one-versus-one” classifiers to a “one-vs-rest” decision function of shape (n_samples, n_classes).

>>> X = [[0], [1], [2], [3]]
>>> Y = [0, 1, 2, 3]
>>> clf = svm.SVC(decision_function_shape='ovo')
>>> clf.fit(X, Y)
SVC(decision_function_shape='ovo')
>>> dec = clf.decision_function([[1]])
>>> dec.shape[1] # 4 classes: 4*3/2 = 6
6
>>> clf.decision_function_shape = "ovr"
>>> dec = clf.decision_function([[1]])
>>> dec.shape[1] # 4 classes
4

On the other hand, LinearSVC implements “one-vs-the-rest” multi-class strategy, thus training n_classes models.

>>> lin_clf = svm.LinearSVC()
>>> lin_clf.fit(X, Y)
LinearSVC()
>>> dec = lin_clf.decision_function([[1]])
>>> dec.shape[1]
4

See Mathematical formulation for a complete description of the decision function.

Note that the LinearSVC also implements an alternative multi-class strategy, the so-called multi-class SVM formulated by Crammer and Singer 16, by using the option multi_class='crammer_singer'. In practice, one-vs-rest classification is usually preferred, since the results are mostly similar, but the runtime is significantly less.

For “one-vs-rest” LinearSVC the attributes coef_ and intercept_ have the shape (n_classes, n_features) and (n_classes,) respectively. Each row of the coefficients corresponds to one of the n_classes “one-vs-rest” classifiers and similar for the intercepts, in the order of the “one” class.

In the case of “one-vs-one” SVC and NuSVC, the layout of the attributes is a little more involved. In the case of a linear kernel, the attributes coef_ and intercept_ have the shape (n_classes * (n_classes - 1) / 2, n_features) and (n_classes * (n_classes - 1) / 2) respectively. This is similar to the layout for LinearSVC described above, with each row now corresponding to a binary classifier. The order for classes 0 to n is “0 vs 1”, “0 vs 2” , … “0 vs n”, “1 vs 2”, “1 vs 3”, “1 vs n”, . . . “n-1 vs n”.

The shape of dual_coef_ is (n_classes-1, n_SV) with a somewhat hard to grasp layout. The columns correspond to the support vectors involved in any of the n_classes * (n_classes - 1) / 2 “one-vs-one” classifiers. Each of the support vectors is used in n_classes - 1 classifiers. The n_classes - 1 entries in each row correspond to the dual coefficients for these classifiers.

This might be clearer with an example: consider a three class problem with class 0 having three support vectors \(v^{0}_0, v^{1}_0, v^{2}_0\) and class 1 and 2 having two support vectors \(v^{0}_1, v^{1}_1\) and \(v^{0}_2, v^{1}_2\) respectively. For each support vector \(v^{j}_i\), there are two dual coefficients. Let’s call the coefficient of support vector \(v^{j}_i\) in the classifier between classes \(i\) and \(k\) \(\alpha^{j}_{i,k}\). Then dual_coef_ looks like this:

1.4.1.2. Scores and probabilities¶

The decision_function method of SVC and NuSVC gives per-class scores for each sample (or a single score per sample in the binary case). When the constructor option probability is set to True, class membership probability estimates (from the methods predict_proba and predict_log_proba) are enabled. In the binary case, the probabilities are calibrated using Platt scaling 9: logistic regression on the SVM’s scores, fit by an additional cross-validation on the training data. In the multiclass case, this is extended as per 10.

The cross-validation involved in Platt scaling is an expensive operation for large datasets. In addition, the probability estimates may be inconsistent with the scores:

• the “argmax” of the scores may not be the argmax of the probabilities

• in binary classification, a sample may be labeled by predict as belonging to the positive class even if the output of predict_proba is less than 0.5; and similarly, it could be labeled as negative even if the output of predict_proba is more than 0.5.

Platt’s method is also known to have theoretical issues. If confidence scores are required, but these do not have to be probabilities, then it is advisable to set probability=False and use decision_function instead of predict_proba.

Please note that when decision_function_shape='ovr' and n_classes > 2, unlike decision_function, the predict method does not try to break ties by default. You can set break_ties=True for the output of predict to be the same as np.argmax(clf.decision_function(...), axis=1), otherwise the first class among the tied classes will always be returned; but have in mind that it comes with a computational cost. See SVM Tie Breaking Example for an example on tie breaking.

1.4.1.3. Unbalanced problems¶

In problems where it is desired to give more importance to certain classes or certain individual samples, the parameters class_weight and sample_weight can be used.

SVC (but not NuSVC) implements the parameter class_weight in the fit method. It’s a dictionary of the form {class_label : value}, where value is a floating point number > 0 that sets the parameter C of class class_label to C * value. The figure below illustrates the decision boundary of an unbalanced problem, with and without weight correction.

SVC, NuSVC, SVR, NuSVR, LinearSVC, LinearSVR and OneClassSVM implement also weights for individual samples in the fit method through the sample_weight parameter. Similar to class_weight, this sets the parameter C for the i-th example to C * sample_weight[i], which will encourage the classifier to get these samples right. The figure below illustrates the effect of sample weighting on the decision boundary. The size of the circles is proportional to the sample weights:

1.4.6.1. Parameters of the RBF Kernel¶

When training an SVM with the Radial Basis Function (RBF) kernel, two parameters must be considered: C and gamma. The parameter C, common to all SVM kernels, trades off misclassification of training examples against simplicity of the decision surface. A low C makes the decision surface smooth, while a high C aims at classifying all training examples correctly. gamma defines how much influence a single training example has. The larger gamma is, the closer other examples must be to be affected.

Proper choice of C and gamma is critical to the SVM’s performance. One is advised to use GridSearchCV with C and gamma spaced exponentially far apart to choose good values.

1.4.6.2. Custom Kernels¶

You can define your own kernels by either giving the kernel as a python function or by precomputing the Gram matrix.

Classifiers with custom kernels behave the same way as any other classifiers, except that:

• Field support_vectors_ is now empty, only indices of support vectors are stored in support_

• A reference (and not a copy) of the first argument in the fit() method is stored for future reference. If that array changes between the use of fit() and predict() you will have unexpected results.

>>> import numpy as np
>>> from sklearn import svm
>>> def my_kernel(X, Y):
...     return np.dot(X, Y.T)
...
>>> clf = svm.SVC(kernel=my_kernel)

>>> import numpy as np
>>> from sklearn.datasets import make_classification
>>> from sklearn.model_selection import train_test_split
>>> from sklearn import svm
>>> X, y = make_classification(n_samples=10, random_state=0)
>>> X_train , X_test , y_train, y_test = train_test_split(X, y, random_state=0)
>>> clf = svm.SVC(kernel='precomputed')
>>> # linear kernel computation
>>> gram_train = np.dot(X_train, X_train.T)
>>> clf.fit(gram_train, y_train)
SVC(kernel='precomputed')
>>> # predict on training examples
>>> gram_test = np.dot(X_test, X_train.T)
>>> clf.predict(gram_test)
array([0, 1, 0])

1.4.7.1. SVC¶

Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, in two classes, and a vector \(y \in \{1, -1\}^n\), our goal is to find \(w \in \mathbb{R}^p\) and \(b \in \mathbb{R}\) such that the prediction given by \(\text{sign} (w^T\phi(x) + b)\) is correct for most samples.

SVC solves the following primal problem:

Intuitively, we’re trying to maximize the margin (by minimizing \(||w||^2 = w^Tw\)), while incurring a penalty when a sample is misclassified or within the margin boundary. Ideally, the value \(y_i (w^T \phi (x_i) + b)\) would be \(\geq 1\) for all samples, which indicates a perfect prediction. But problems are usually not always perfectly separable with a hyperplane, so we allow some samples to be at a distance \(\zeta_i\) from their correct margin boundary. The penalty term C controls the strengh of this penalty, and as a result, acts as an inverse regularization parameter (see note below).

The dual problem to the primal is

where \(e\) is the vector of all ones, and \(Q\) is an \(n\) by \(n\) positive semidefinite matrix, \(Q_{ij} \equiv y_i y_j K(x_i, x_j)\), where \(K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\) is the kernel. The terms \(\alpha_i\) are called the dual coefficients, and they are upper-bounded by \(C\). This dual representation highlights the fact that training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function \(\phi\): see kernel trick.

Once the optimization problem is solved, the output of decision_function for a given sample \(x\) becomes:

and the predicted class correspond to its sign. We only need to sum over the support vectors (i.e. the samples that lie within the margin) because the dual coefficients \(\alpha_i\) are zero for the other samples.

These parameters can be accessed through the attributes dual_coef_ which holds the product \(y_i \alpha_i\), support_vectors_ which holds the support vectors, and intercept_ which holds the independent term \(b\)

1.4.7.2. LinearSVC¶

The primal problem can be equivalently formulated as

where we make use of the hinge loss. This is the form that is directly optimized by LinearSVC, but unlike the dual form, this one does not involve inner products between samples, so the famous kernel trick cannot be applied. This is why only the linear kernel is supported by LinearSVC (\(\phi\) is the identity function).

1.4.7.3. NuSVC¶

The \(\nu\)-SVC formulation 15 is a reparameterization of the \(C\)-SVC and therefore mathematically equivalent.

We introduce a new parameter \(\nu\) (instead of \(C\)) which controls the number of support vectors and margin errors: \(\nu \in (0, 1]\) is an upper bound on the fraction of margin errors and a lower bound of the fraction of support vectors. A margin error corresponds to a sample that lies on the wrong side of its margin boundary: it is either misclassified, or it is correctly classified but does not lie beyond the margin.

1.4.7.4. SVR¶

Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, and a vector \(y \in \mathbb{R}^n\) \(\varepsilon\)-SVR solves the following primal problem:

Here, we are penalizing samples whose prediction is at least \(\varepsilon\) away from their true target. These samples penalize the objective by \(\zeta_i\) or \(\zeta_i^*\), depending on whether their predictions lie above or below the \(\varepsilon\) tube.

The dual problem is

where \(e\) is the vector of all ones, \(Q\) is an \(n\) by \(n\) positive semidefinite matrix, \(Q_{ij} \equiv K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\) is the kernel. Here training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function \(\phi\).

The prediction is:

These parameters can be accessed through the attributes dual_coef_ which holds the difference \(\alpha_i - \alpha_i^*\), support_vectors_ which holds the support vectors, and intercept_ which holds the independent term \(b\)

1.4.7.5. LinearSVR¶

The primal problem can be equivalently formulated as

where we make use of the epsilon-insensitive loss, i.e. errors of less than \(\varepsilon\) are ignored. This is the form that is directly optimized by LinearSVR.