View source for Regularization perspectives on support-vector machines

Within [[Mathematical analysis|mathematical analysis]], '''Regularization perspectives on support-vector machines''' provide a way of interpreting [[Support-vector machine|support-vector machine]]s (SVMs) in the context of other regularization-based machine-learning algorithms.  SVM algorithms categorize binary data, with the goal of fitting the training set data in a way that minimizes the average of the hinge-loss function and L2 norm of the learned weights. This strategy avoids [[Overfitting|overfitting]] via [[Tikhonov regularization]] and in the L2 norm sense and also corresponds to minimizing the bias and variance of our estimator of the weights. Estimators with lower [[Mean squared error]] predict better or generalize better when given unseen data.
Specifically, [[Tikhonov regularization]] algorithms produce a decision boundary that minimizes the average training-set error and constrain the [[Decision boundary]] not to be excessively complicated or overfit the training data via a L2 norm of the weights term. The training and test-set errors can be measured without bias and in a fair way using accuracy, precision, Auc-Roc, precision-recall, and other metrics. 

Regularization perspectives on support-vector machines interpret SVM as a special case of Tikhonov regularization, specifically Tikhonov regularization with the [[Hinge loss|hinge loss]] for a loss function.  This provides a theoretical framework with which to analyze SVM algorithms and compare them to other algorithms with the same goals: to generalize without [[Overfitting|overfitting]]. SVM was first proposed in 1995 by [[Biography:Corinna Cortes|Corinna Cortes]] and [[Biography:Vladimir Vapnik|Vladimir Vapnik]], and framed geometrically as a method for finding [[Hyperplane|hyperplane]]s that can separate multidimensional data into two categories.<ref>{{cite journal |last=Cortes |first=Corinna |author2=Vladimir Vapnik |title=Support-Vector Networks |journal=Machine Learning |year=1995 |volume=20 |issue=3 |pages=273–297 |doi=10.1007/BF00994018 |doi-access=free }}</ref> This traditional geometric interpretation of SVMs provides useful intuition about how SVMs work, but is difficult to relate to other machine-learning techniques for avoiding overfitting, like [[Regularization (mathematics)|regularization]], [[Early stopping|early stopping]], sparsity and [[Bayesian inference]].  However, once it was discovered that SVM is also a [[Special case|special case]] of Tikhonov regularization, regularization perspectives on SVM provided the theory necessary to fit SVM within a broader class of algorithms.<ref name="rosasco1">{{cite web |last=Rosasco |first=Lorenzo |title=Regularized Least-Squares and Support Vector Machines |url=https://www.mit.edu/~9.520/spring12/slides/class06/class06_RLSSVM.pdf}}</ref><ref>{{cite book |last=Rifkin |first=Ryan |title=Everything Old is New Again: A Fresh Look at Historical Approaches in Machine Learning |year=2002 |publisher=MIT (PhD thesis) |url=http://web.mit.edu/~9.520/www/Papers/thesis-rifkin.pdf}}</ref><ref name="Lee 2012 67–81">{{cite journal |last1=Lee |first1=Yoonkyung |first2=Grace |last2=Wahba |title=Multicategory Support Vector Machines |journal=Journal of the American Statistical Association |year=2012 |volume=99 |issue=465 |pages=67–81 |doi=10.1198/016214504000000098 }}</ref> This has enabled detailed comparisons between SVM and other forms of Tikhonov regularization, and theoretical grounding for why it is beneficial to use SVM's loss function, the hinge loss.<ref name="Rosasco 2004 1063–1076">{{cite journal |authors=Rosasco L., De Vito E., Caponnetto A., Piana M., Verri A. |title=Are Loss Functions All the Same |journal=Neural Computation |date=May 2004 |volume=16 |issue=5 |series=5 |pages=1063–1076 |doi=10.1162/089976604773135104 |pmid=15070510|citeseerx=10.1.1.109.6786 }}</ref>

==Theoretical background==
In the [[Statistical learning theory|statistical learning theory]] framework, an [[Algorithm|algorithm]] is a strategy for choosing a [[Function (mathematics)|function]] <math>f \colon \mathbf X \to \mathbf Y </math> given a training set <math>S = \{(x_1, y_1), \ldots, (x_n, y_n)\}</math> of inputs <math>x_i</math> and their labels <math>y_i</math> (the labels are usually <math>\pm1</math>).  [[Regularization (mathematics)|Regularization]] strategies avoid [[Overfitting|overfitting]] by choosing a function that fits the data, but is not too complex.  Specifically:

: <math>f = \underset{f\in\mathcal{H}} \operatorname{argmin}\left\{\frac 1 n \sum_{i=1}^n V(y_i, f(x_i)) + \lambda\|f\|^2_\mathcal{H}\right\},</math>

where <math>\mathcal{H}</math> is a hypothesis space<ref>A hypothesis space is the set of functions used to model the data in a machine-learning problem. Each function corresponds to a hypothesis about the structure of the data. Typically the functions in a hypothesis space form a [[Hilbert space]] of functions with norm formed from the loss function.</ref> of functions, <math>V \colon \mathbf Y \times \mathbf Y \to \mathbb R</math> is the loss function, <math>\|\cdot\|_\mathcal H</math> is a [[Norm (mathematics)|norm]] on the hypothesis space of functions, and <math>\lambda \in \mathbb R</math> is the regularization parameter.<ref>For insight on choosing the parameter, see, e.g., {{cite journal |last=Wahba |first=Grace |author2=Yonghua Wang |title=When is the optimal regularization parameter insensitive to the choice of the loss function |journal=Communications in Statistics – Theory and Methods |year=1990 |volume=19 |issue=5 |pages=1685–1700 |doi=10.1080/03610929008830285 }}</ref>

When <math>\mathcal{H}</math> is a [[Reproducing kernel Hilbert space|reproducing kernel Hilbert space]], there exists a kernel function <math>K \colon \mathbf X \times \mathbf X \to \mathbb R</math> that can be written as an <math>n \times n</math> symmetric [[Positive-definite kernel|positive-definite]] [[Matrix (mathematics)|matrix]] <math>\mathbf K</math>.  By the [[Representer theorem|representer theorem]],<ref>See {{cite book |last=Scholkopf |first=Bernhard |author2=Ralf Herbrich |author3=Alex Smola |title=A Generalized Representer Theorem |journal=Computational Learning Theory: Lecture Notes in Computer Science |year=2001 |volume=2111 |pages=416–426 |doi=10.1007/3-540-44581-1_27 |series=Lecture Notes in Computer Science |isbn=978-3-540-42343-0 |citeseerx=10.1.1.42.8617 }}</ref>
: <math>f(x_i) = \sum_{j=1}^n c_j \mathbf K_{ij}, \text{ and } \|f\|^2_{\mathcal H} = \langle f, f\rangle_\mathcal H = \sum_{i=1}^n \sum_{j=1}^n c_i c_jK(x_i, x_j) = c^T \mathbf K c.</math>

==Special properties of the hinge loss==
[[File:Hinge and Misclassification Loss.png|Hinge and misclassification loss functions]]

The simplest and most intuitive loss function for categorization is the misclassification loss, or 0–1 loss, which is 0 if <math>f(x_i) = y_i</math> and 1 if <math>f(x_i) \neq y_i</math>, i.e. the [[Heaviside step function]] on <math>-y_if(x_i)</math>.  However, this loss function is not [[Convex function|convex]], which makes the regularization problem very difficult to minimize computationally.  Therefore, we look for convex substitutes for the 0–1 loss.  The hinge loss, <math>V\big(y_i, f(x_i)\big) = \big(1 - yf(x)\big)_+</math>, where <math>(s)_+ = \max(s, 0)</math>, provides such a convex relaxation.  In fact, the hinge loss is the tightest convex upper bound to the 0–1 misclassification loss function,<ref name="Lee 2012 67–81"/> and with infinite data returns the [[Bayes' theorem|Bayes]]-optimal solution:<ref name="Rosasco 2004 1063–1076"/><ref>{{cite journal |last=Lin |first=Yi |title=Support Vector Machines and the Bayes Rule in Classification |journal=Data Mining and Knowledge Discovery |date=July 2002 |volume=6 |issue=3 |pages=259–275 |doi=10.1023/A:1015469627679 |url=http://cbio.ensmp.fr/~jvert/svn/bibli/local/Lin2002Support.pdf}}</ref>

: <math>f_b(x) = \begin{cases}
  1, & p(1\mid x) > p(-1\mid x), \\
 -1, & p(1\mid x) < p(-1\mid x).
\end{cases}</math>

==Derivation==

The Tikhonov regularization problem can be shown to be equivalent to traditional formulations of SVM by expressing it in terms of the hinge loss.<ref>For a detailed derivation, see {{cite book |last=Rifkin |first=Ryan |title=Everything Old is New Again: A Fresh Look at Historical Approaches in Machine Learning |year=2002 |publisher=MIT (PhD thesis) |url=http://web.mit.edu/~9.520/www/Papers/thesis-rifkin.pdf}}</ref> With the hinge loss

: <math>V\big(y_i, f(x_i)\big) = \big(1 - yf(x)\big)_+,</math>

where <math>(s)_+ = \max(s, 0)</math>, the regularization problem becomes

: <math>f = \underset{f\in\mathcal{H}} \operatorname{argmin} \left\{\frac 1 n \sum_{i=1}^n \big(1 - yf(x)\big)_+ + \lambda\|f\|^2_\mathcal{H}\right\}.</math>

Multiplying by <math>1/(2\lambda)</math> yields

: <math>f = \underset{f\in\mathcal{H}} \operatorname{argmin} \left\{C\sum_{i=1}^n \big(1 - yf(x)\big)_+ + \frac{1}{2}\|f\|^2_\mathcal{H}\right\}</math>

with <math>C = 1/(2\lambda n)</math>, which is equivalent to the standard SVM minimization problem.

==Notes and references==
{{Reflist}}

* {{cite journal|last=Evgeniou|first=Theodoros |author2=Massimiliano Pontil |author3=Tomaso Poggio|title=Regularization Networks and Support Vector Machines|journal=Advances in Computational Mathematics|year=2000|volume=13|issue=1|pages=1–50|doi=10.1023/A:1018946025316|url=http://cbcl.mit.edu/projects/cbcl/publications/ps/evgeniou-reviewall.pdf}}
* {{cite web|last=Joachims|first=Thorsten|title=SVMlight|url=http://svmlight.joachims.org/|access-date=2012-05-18|archive-url=https://web.archive.org/web/20150419060655/http://svmlight.joachims.org/|archive-date=2015-04-19|url-status=dead}}
* {{cite book|last=Vapnik|first=Vladimir|title=The Nature of Statistical Learning Theory|year=1999|publisher=Springer-Verlag|location=New York|isbn=978-0-387-98780-4|url=https://books.google.com/books?id=sna9BaxVbj8C&q=vapnik+the+nature+of+statistical+learning+theory&pg=PR7}}

[[Category:Support vector machines]]
[[Category:Mathematical analysis]]

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