Support vector machine

If the training data is linearly separable, we can select two parallel hyperplanes that separate the two classes of data, so that the distance between them is as large as possible. The region bounded by these two hyperplanes is called the "margin", and the maximum-margin hyperplane is the hyperplane that lies halfway between them. With a normalized or standardized dataset, these hyperplanes can be described by the equations

${\displaystyle \mathbf {w} ^{\mathsf {T}}\mathbf {x} -b=1}$ (anything on or above this boundary is of one class, with label 1)

and

${\displaystyle \mathbf {w} ^{\mathsf {T}}\mathbf {x} -b=-1}$ (anything on or below this boundary is of the other class, with label −1).

Geometrically, the distance between these two hyperplanes is ${\displaystyle {\tfrac {2}{\|\mathbf {w} \|}}}$,[17] so to maximize the distance between the planes we want to minimize ${\displaystyle \|\mathbf {w} \|}$. The distance is computed using the distance from a point to a plane equation. We also have to prevent data points from falling into the margin, we add the following constraint: for each ${\displaystyle i}$ either

$${\displaystyle \mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b\geq 1\,,{\text{ if }}y_{i}=1,}$$

or

$${\displaystyle \mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b\leq -1\,,{\text{ if }}y_{i}=-1.}$$

These constraints state that each data point must lie on the correct side of the margin.

This can be rewritten as

$${\displaystyle y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\geq 1,\quad {\text{ for all }}1\leq i\leq n.}$$

(1)

We can put this together to get the optimization problem:

"Minimize ${\displaystyle \|\mathbf {w} \|}$ subject to ${\displaystyle y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\geq 1}$ for ${\displaystyle i=1,\ldots ,n}$."

The ${\displaystyle \mathbf {w} }$ and ${\displaystyle b}$ that solve this problem determine our classifier, ${\displaystyle \mathbf {x} \mapsto \operatorname {sgn}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} -b)}$ where ${\displaystyle \operatorname {sgn}(\cdot )}$ is the sign function.

An important consequence of this geometric description is that the max-margin hyperplane is completely determined by those ${\displaystyle \mathbf {x} _{i}}$ that lie nearest to it. These ${\displaystyle \mathbf {x} _{i}}$ are called support vectors.

To extend SVM to cases in which the data are not linearly separable, the hinge loss function is helpful

$${\displaystyle \max \left(0,1-y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\right).}$$

Note that ${\displaystyle y_{i}}$ is the i-th target (i.e., in this case, 1 or −1), and ${\displaystyle \mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b}$ is the i-th output.

This function is zero if the constraint in (1) is satisfied, in other words, if ${\displaystyle \mathbf {x} _{i}}$ lies on the correct side of the margin. For data on the wrong side of the margin, the function's value is proportional to the distance from the margin.

The goal of the optimization then is to minimize

$${\displaystyle \lambda \lVert \mathbf {w} \rVert ^{2}+\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\right)\right],}$$

where the parameter ${\displaystyle \lambda >0}$ determines the trade-off between increasing the margin size and ensuring that the ${\displaystyle \mathbf {x} _{i}}$ lie on the correct side of the margin. Thus, for sufficiently small values of ${\displaystyle \lambda }$, it will behave similar to the hard-margin SVM, if the input data are linearly classifiable, but will still learn if a classification rule is viable or not. (This parameter ${\displaystyle \lambda }$ is often also called ${\displaystyle C}$, e.g. in LIBSVM, but usually refers to the inverse of ${\displaystyle \lambda }$.)

Minimizing (2) can be rewritten as a constrained optimization problem with a differentiable objective function in the following way.

For each ${\displaystyle i\in \{1,\,\ldots ,\,n\}}$ we introduce a variable ${\displaystyle \zeta _{i}=\max \left(0,1-y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\right)}$. Note that ${\displaystyle \zeta _{i}}$ is the smallest nonnegative number satisfying ${\displaystyle y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\geq 1-\zeta _{i}.}$

Thus we can rewrite the optimization problem as follows

$${\displaystyle {\begin{aligned}&{\text{minimize }}{\frac {1}{n}}\sum _{i=1}^{n}\zeta _{i}+\lambda \|\mathbf {w} \|^{2}\\[0.5ex]&{\text{subject to }}y_{i}\left(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b\right)\geq 1-\zeta _{i}\,{\text{ and }}\,\zeta _{i}\geq 0,\,{\text{for all }}i.\end{aligned}}}$$

This is called the primal problem.

By solving for the Lagrangian dual of the above problem, one obtains the simplified problem

$${\displaystyle {\begin{aligned}&{\text{maximize}}\,\,f(c_{1}\ldots c_{n})=\sum _{i=1}^{n}c_{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}c_{i}(\mathbf {x} _{i}^{\mathsf {T}}\mathbf {x} _{j})y_{j}c_{j},\\&{\text{subject to }}\sum _{i=1}^{n}c_{i}y_{i}=0,\,{\text{and }}0\leq c_{i}\leq {\frac {1}{2n\lambda }}\;{\text{for all }}i.\end{aligned}}}$$

This is called the dual problem. Since the dual maximization problem is a quadratic function of the ${\displaystyle c_{i}}$ subject to linear constraints, it is efficiently solvable by quadratic programming algorithms.

Here, the variables ${\displaystyle c_{i}}$ are defined such that

$${\displaystyle \mathbf {w} =\sum _{i=1}^{n}c_{i}y_{i}\mathbf {x} _{i}.}$$

Moreover, ${\displaystyle c_{i}=0}$ exactly when ${\displaystyle \mathbf {x} _{i}}$ lies on the correct side of the margin, and ${\displaystyle 0<c_{i}<(2n\lambda )^{-1}}$ when ${\displaystyle \mathbf {x} _{i}}$ lies on the margin's boundary. It follows that ${\displaystyle \mathbf {w} }$ can be written as a linear combination of the support vectors.

The offset, ${\displaystyle b}$, can be recovered by finding an ${\displaystyle \mathbf {x} _{i}}$ on the margin's boundary and solving

$${\displaystyle y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)=1\iff b=\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-y_{i}.}$$

(Note that ${\displaystyle y_{i}^{-1}=y_{i}}$ since ${\displaystyle y_{i}=\pm 1}$.)

A training example of SVM with kernel given by φ((a, b)) = (a, b, a² + b²)

Suppose now that we would like to learn a nonlinear classification rule which corresponds to a linear classification rule for the transformed data points ${\displaystyle \varphi (\mathbf {x} _{i}).}$ Moreover, we are given a kernel function ${\displaystyle k}$ which satisfies ${\displaystyle k(\mathbf {x} _{i},\mathbf {x} _{j})=\varphi (\mathbf {x} _{i})\cdot \varphi (\mathbf {x} _{j})}$.

We know the classification vector ${\displaystyle \mathbf {w} }$ in the transformed space satisfies

$${\displaystyle \mathbf {w} =\sum _{i=1}^{n}c_{i}y_{i}\varphi (\mathbf {x} _{i}),}$$

where, the ${\displaystyle c_{i}}$ are obtained by solving the optimization problem

$${\displaystyle {\begin{aligned}{\text{maximize}}\,\,f(c_{1}\ldots c_{n})&=\sum _{i=1}^{n}c_{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}c_{i}(\varphi (\mathbf {x} _{i})\cdot \varphi (\mathbf {x} _{j}))y_{j}c_{j}\\&=\sum _{i=1}^{n}c_{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}c_{i}k(\mathbf {x} _{i},\mathbf {x} _{j})y_{j}c_{j}\\{\text{subject to }}\sum _{i=1}^{n}c_{i}y_{i}&=0,\,{\text{and }}0\leq c_{i}\leq {\frac {1}{2n\lambda }}\;{\text{for all }}i.\end{aligned}}}$$

The coefficients ${\displaystyle c_{i}}$ can be solved for using quadratic programming, as before. Again, we can find some index ${\displaystyle i}$ such that ${\displaystyle 0<c_{i}<(2n\lambda )^{-1}}$, so that ${\displaystyle \varphi (\mathbf {x} _{i})}$ lies on the boundary of the margin in the transformed space, and then solve

$${\displaystyle {\begin{aligned}b=\mathbf {w} ^{\mathsf {T}}\varphi (\mathbf {x} _{i})-y_{i}&=\left[\sum _{j=1}^{n}c_{j}y_{j}\varphi (\mathbf {x} _{j})\cdot \varphi (\mathbf {x} _{i})\right]-y_{i}\\&=\left[\sum _{j=1}^{n}c_{j}y_{j}k(\mathbf {x} _{j},\mathbf {x} _{i})\right]-y_{i}.\end{aligned}}}$$

Finally,

$${\displaystyle \mathbf {z} \mapsto \operatorname {sgn}(\mathbf {w} ^{\mathsf {T}}\varphi (\mathbf {z} )-b)=\operatorname {sgn} \left(\left[\sum _{i=1}^{n}c_{i}y_{i}k(\mathbf {x} _{i},\mathbf {z} )\right]-b\right).}$$

Recent algorithms for finding the SVM classifier include sub-gradient descent and coordinate descent. Both techniques have proven to offer significant advantages over the traditional approach when dealing with large, sparse datasets—sub-gradient methods are especially efficient when there are many training examples, and coordinate descent when the dimension of the feature space is high.

Sub-gradient descent algorithms for the SVM work directly with the expression

$${\displaystyle f(\mathbf {w} ,b)=\left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i}-b)\right)\right]+\lambda \|\mathbf {w} \|^{2}.}$$

Note that ${\displaystyle f}$ is a convex function of ${\displaystyle \mathbf {w} }$ and ${\displaystyle b}$. As such, traditional gradient descent (or SGD) methods can be adapted, where instead of taking a step in the direction of the function's gradient, a step is taken in the direction of a vector selected from the function's sub-gradient. This approach has the advantage that, for certain implementations, the number of iterations does not scale with ${\displaystyle n}$, the number of data points.[20]

Coordinate descent algorithms for the SVM work from the dual problem

$${\displaystyle {\begin{aligned}&{\text{maximize}}\,\,f(c_{1}\ldots c_{n})=\sum _{i=1}^{n}c_{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}c_{i}(x_{i}\cdot x_{j})y_{j}c_{j},\\&{\text{subject to }}\sum _{i=1}^{n}c_{i}y_{i}=0,\,{\text{and }}0\leq c_{i}\leq {\frac {1}{2n\lambda }}\;{\text{for all }}i.\end{aligned}}}$$

For each ${\displaystyle i\in \{1,\,\ldots ,\,n\}}$, iteratively, the coefficient ${\displaystyle c_{i}}$ is adjusted in the direction of ${\displaystyle \partial f/\partial c_{i}}$. Then, the resulting vector of coefficients ${\displaystyle (c_{1}',\,\ldots ,\,c_{n}')}$ is projected onto the nearest vector of coefficients that satisfies the given constraints. (Typically Euclidean distances are used.) The process is then repeated until a near-optimal vector of coefficients is obtained. The resulting algorithm is extremely fast in practice, although few performance guarantees have been proven.[21]

In supervised learning, one is given a set of training examples ${\displaystyle X_{1}\ldots X_{n}}$ with labels ${\displaystyle y_{1}\ldots y_{n}}$, and wishes to predict ${\displaystyle y_{n+1}}$ given ${\displaystyle X_{n+1}}$. To do so one forms a hypothesis, ${\displaystyle f}$, such that ${\displaystyle f(X_{n+1})}$ is a "good" approximation of ${\displaystyle y_{n+1}}$. A "good" approximation is usually defined with the help of a loss function, ${\displaystyle \ell (y,z)}$, which characterizes how bad ${\displaystyle z}$ is as a prediction of ${\displaystyle y}$. We would then like to choose a hypothesis that minimizes the expected risk:

$${\displaystyle \varepsilon (f)=\mathbb {E} \left[\ell (y_{n+1},f(X_{n+1}))\right].}$$

In most cases, we don't know the joint distribution of ${\displaystyle X_{n+1},\,y_{n+1}}$ outright. In these cases, a common strategy is to choose the hypothesis that minimizes the empirical risk:

$${\displaystyle {\hat {\varepsilon }}(f)={\frac {1}{n}}\sum _{k=1}^{n}\ell (y_{k},f(X_{k})).}$$

Under certain assumptions about the sequence of random variables ${\displaystyle X_{k},\,y_{k}}$ (for example, that they are generated by a finite Markov process), if the set of hypotheses being considered is small enough, the minimizer of the empirical risk will closely approximate the minimizer of the expected risk as ${\displaystyle n}$ grows large. This approach is called empirical risk minimization, or ERM.

In order for the minimization problem to have a well-defined solution, we have to place constraints on the set ${\displaystyle {\mathcal {H}}}$ of hypotheses being considered. If ${\displaystyle {\mathcal {H}}}$ is a normed space (as is the case for SVM), a particularly effective technique is to consider only those hypotheses ${\displaystyle f}$ for which ${\displaystyle \lVert f\rVert _{\mathcal {H}}<k}$ . This is equivalent to imposing a regularization penalty ${\displaystyle {\mathcal {R}}(f)=\lambda _{k}\lVert f\rVert _{\mathcal {H}}}$, and solving the new optimization problem

$${\displaystyle {\hat {f}}=\mathrm {arg} \min _{f\in {\mathcal {H}}}{\hat {\varepsilon }}(f)+{\mathcal {R}}(f).}$$

This approach is called Tikhonov regularization.

More generally, ${\displaystyle {\mathcal {R}}(f)}$ can be some measure of the complexity of the hypothesis ${\displaystyle f}$, so that simpler hypotheses are preferred.

Recall that the (soft-margin) SVM classifier ${\displaystyle {\hat {\mathbf {w} }},b:\mathbf {x} \mapsto \operatorname {sgn}({\hat {\mathbf {w} }}^{\mathsf {T}}\mathbf {x} -b)}$ is chosen to minimize the following expression:

$${\displaystyle \left[{\frac {1}{n}}\sum _{i=1}^{n}\max \left(0,1-y_{i}(\mathbf {w} ^{\mathsf {T}}\mathbf {x} -b)\right)\right]+\lambda \|\mathbf {w} \|^{2}.}$$

In light of the above discussion, we see that the SVM technique is equivalent to empirical risk minimization with Tikhonov regularization, where in this case the loss function is the hinge loss

$${\displaystyle \ell (y,z)=\max \left(0,1-yz\right).}$$

From this perspective, SVM is closely related to other fundamental classification algorithms such as regularized least-squares and logistic regression. The difference between the three lies in the choice of loss function: regularized least-squares amounts to empirical risk minimization with the square-loss, ${\displaystyle \ell _{sq}(y,z)=(y-z)^{2}}$; logistic regression employs the log-loss,

$${\displaystyle \ell _{\log }(y,z)=\ln(1+e^{-yz}).}$$

The difference between the hinge loss and these other loss functions is best stated in terms of target functions - the function that minimizes expected risk for a given pair of random variables ${\displaystyle X,\,y}$.

In particular, let ${\displaystyle y_{x}}$ denote ${\displaystyle y}$ conditional on the event that ${\displaystyle X=x}$. In the classification setting, we have:

$${\displaystyle y_{x}={\begin{cases}1&{\text{with probability }}p_{x}\\-1&{\text{with probability }}1-p_{x}\end{cases}}}$$

The optimal classifier is therefore:

$${\displaystyle f^{*}(x)={\begin{cases}1&{\text{if }}p_{x}\geq 1/2\\-1&{\text{otherwise}}\end{cases}}}$$

For the square-loss, the target function is the conditional expectation function, ${\displaystyle f_{sq}(x)=\mathbb {E} \left[y_{x}\right]}$; For the logistic loss, it's the logit function, ${\displaystyle f_{\log }(x)=\ln \left(p_{x}/({1-p_{x}})\right)}$. While both of these target functions yield the correct classifier, as ${\displaystyle \operatorname {sgn}(f_{sq})=\operatorname {sgn}(f_{\log })=f^{*}}$, they give us more information than we need. In fact, they give us enough information to completely describe the distribution of ${\displaystyle y_{x}}$.

On the other hand, one can check that the target function for the hinge loss is exactly ${\displaystyle f^{*}}$. Thus, in a sufficiently rich hypothesis space—or equivalently, for an appropriately chosen kernel—the SVM classifier will converge to the simplest function (in terms of ${\displaystyle {\mathcal {R}}}$) that correctly classifies the data. This extends the geometric interpretation of SVM—for linear classification, the empirical risk is minimized by any function whose margins lie between the support vectors, and the simplest of these is the max-margin classifier.[22]

The effectiveness of SVM depends on the selection of kernel, the kernel's parameters, and soft margin parameter ${\displaystyle \lambda }$. A common choice is a Gaussian kernel, which has a single parameter ${\displaystyle \gamma }$. The best combination of ${\displaystyle \lambda }$ and ${\displaystyle \gamma }$ is often selected by a grid search with exponentially growing sequences of ${\displaystyle \lambda }$ and ${\displaystyle \gamma }$, for example, ${\displaystyle \lambda \in \{2^{-5},2^{-3},\dots ,2^{13},2^{15}\}}$; ${\displaystyle \gamma \in \{2^{-15},2^{-13},\dots ,2^{1},2^{3}\}}$. Typically, each combination of parameter choices is checked using cross validation, and the parameters with best cross-validation accuracy are picked. Alternatively, recent work in Bayesian optimization can be used to select ${\displaystyle \lambda }$ and ${\displaystyle \gamma }$ , often requiring the evaluation of far fewer parameter combinations than grid search. The final model, which is used for testing and for classifying new data, is then trained on the whole training set using the selected parameters.[24]

Potential drawbacks of the SVM include the following aspects:

• Requires full labeling of input data
• Uncalibrated class membership probabilities—SVM stems from Vapnik's theory which avoids estimating probabilities on finite data
• The SVM is only directly applicable for two-class tasks. Therefore, algorithms that reduce the multi-class task to several binary problems have to be applied; see the multi-class SVM section.
• Parameters of a solved model are difficult to interpret.

SVC is a similar method that also builds on kernel functions but is appropriate for unsupervised learning.

Multiclass SVM aims to assign labels to instances by using support vector machines, where the labels are drawn from a finite set of several elements.

The dominant approach for doing so is to reduce the single multiclass problem into multiple binary classification problems.[25] Common methods for such reduction include:[25][26]

• Building binary classifiers that distinguish between one of the labels and the rest (one-versus-all) or between every pair of classes (one-versus-one). Classification of new instances for the one-versus-all case is done by a winner-takes-all strategy, in which the classifier with the highest-output function assigns the class (it is important that the output functions be calibrated to produce comparable scores). For the one-versus-one approach, classification is done by a max-wins voting strategy, in which every classifier assigns the instance to one of the two classes, then the vote for the assigned class is increased by one vote, and finally the class with the most votes determines the instance classification.
• Directed acyclic graph SVM (DAGSVM)[27]
• Error-correcting output codes[28]

Crammer and Singer proposed a multiclass SVM method which casts the multiclass classification problem into a single optimization problem, rather than decomposing it into multiple binary classification problems.[29] See also Lee, Lin and Wahba[30][31] and Van den Burg and Groenen.[32]

Transductive support vector machines extend SVMs in that they could also treat partially labeled data in semi-supervised learning by following the principles of transduction. Here, in addition to the training set ${\displaystyle {\mathcal {D}}}$, the learner is also given a set

$${\displaystyle {\mathcal {D}}^{\star }=\{\mathbf {x} _{i}^{\star }\mid \mathbf {x} _{i}^{\star }\in \mathbb {R} ^{p}\}_{i=1}^{k}}$$

of test examples to be classified. Formally, a transductive support vector machine is defined by the following primal optimization problem:[33]

Minimize (in ${\displaystyle \mathbf {w} ,b,\mathbf {y} ^{\star }}$)

$${\displaystyle {\frac {1}{2}}\|\mathbf {w} \|^{2}}$$

subject to (for any ${\displaystyle i=1,\dots ,n}$ and any ${\displaystyle j=1,\dots ,k}$)

$${\displaystyle {\begin{aligned}&y_{i}(\mathbf {w} \cdot \mathbf {x} _{i}-b)\geq 1,\\&y_{j}^{\star }(\mathbf {w} \cdot \mathbf {x} _{j}^{\star }-b)\geq 1,\end{aligned}}}$$

and

$${\displaystyle y_{j}^{\star }\in \{-1,1\}.}$$

Transductive support vector machines were introduced by Vladimir N. Vapnik in 1998.

SVMs have been generalized to structured SVMs, where the label space is structured and of possibly infinite size.

Support vector regression (prediction) with different thresholds ε. As ε increases, the prediction becomes less sensitive to errors.

A version of SVM for regression was proposed in 1996 by Vladimir N. Vapnik, Harris Drucker, Christopher J. C. Burges, Linda Kaufman and Alexander J. Smola.[34] This method is called support vector regression (SVR). The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by SVR depends only on a subset of the training data, because the cost function for building the model ignores any training data close to the model prediction. Another SVM version known as least-squares support vector machine (LS-SVM) has been proposed by Suykens and Vandewalle.[35]

Training the original SVR means solving[36]

minimize ${\displaystyle {\tfrac {1}{2}}\|w\|^{2}}$

subject to ${\displaystyle |y_{i}-\langle w,x_{i}\rangle -b|\leq \varepsilon }$

where ${\displaystyle x_{i}}$ is a training sample with target value ${\displaystyle y_{i}}$. The inner product plus intercept ${\displaystyle \langle w,x_{i}\rangle +b}$ is the prediction for that sample, and ${\displaystyle \varepsilon }$ is a free parameter that serves as a threshold: all predictions have to be within an ${\displaystyle \varepsilon }$ range of the true predictions. Slack variables are usually added into the above to allow for errors and to allow approximation in the case the above problem is infeasible.

In 2011 it was shown by Polson and Scott that the SVM admits a Bayesian interpretation through the technique of data augmentation.[37] In this approach the SVM is viewed as a graphical model (where the parameters are connected via probability distributions). This extended view allows the application of Bayesian techniques to SVMs, such as flexible feature modeling, automatic hyperparameter tuning, and predictive uncertainty quantification. Recently, a scalable version of the Bayesian SVM was developed by Florian Wenzel, enabling the application of Bayesian SVMs to big data.[38] Florian Wenzel developed two different versions, a variational inference (VI) scheme for the Bayesian kernel support vector machine (SVM) and a stochastic version (SVI) for the linear Bayesian SVM.[39]