Regularization perspectives on support vector machines: Difference between revisions

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'''Regularization perspectives on support vector machines''' provide a way of interpreting [[support vector machine]]s (SVMs) in the context of other machine learning algorithms. SVM algorithms categorize [[multidimensional]] data, with the goal of fitting the [[training set]] data well, but also avoiding [[overfitting]], so that the solution [[generalize]]s to new data points. [[Regularization (mathematics)|Regularization]] algorithms also aim to fit training set data and avoid overfitting. They do this by choosing a fitting function that has low error on the training set, but also is not too complicated, where complicated functions are functions with high [[norm (mathematics)|norm]]s in some [[function space]]. Specifically, [[Tikhonov regularization]] algorithms choose a function that minimize the sum of training set error plus the function's norm. The training set error can be calculated with different [[loss function]]s. For example, [[regularized least squares]] is a special case of Tikhonov regularization using the [[squared error loss]] as the loss function.<ref name="rosasco1"> {{cite web|last=Rosasco|first=Lorenzo|title=Regularized Least-Squares and Support Vector Machines|url=http://www.mit.edu/~9.520/spring12/slides/class06/class06_RLSSVM.pdf}}

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Regularization perspectives on support vector machines interpret SVM as a special case Tikhonov regularization, specifically Tikhonov regularization with the [[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]]. SVM was first proposed in 1995 by [[Corinna Cortes]] and [[Vladimir Vapnik]], and framed geometrically as a method for finding [[hyperplane]]s that can separate [[multidimensional]] data into two categories.<ref>{{cite journal|last=Cortes|first=Corinna|coauthors=Vladimir Vapnik|title=Suppor-Vector Networks|journal=Machine Learning|year=1995|volume=20|pages=273–297|doi=10.1007/BF00994018|url=http://www.springerlink.com/content/k238jx04hm87j80g/?MUD=MP}}</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]], [[sparsity]] and [[Bayesian inference]]. However, once it was discovered that SVM is also a [[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"/><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}}

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==Theoretical background==

In the [[statistical learning theory]] framework, an [[algorithm]] is a strategy for choosing a [[function (mathematics)|function]] <math> f:\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]] by choosing a function that fits the data, but is not too complex. Specifically:

<math>f = \text{arg}\min_{f\in\mathcal{H}}\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:\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|coauthors=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|url=http://www.tandfonline.com/doi/abs/10.1080/03610929008830285}}</ref>

When <math>\mathcal{H}</math> is a [[reproducing kernel Hilbert space]], there exists a [[kernel function]] <math>K: \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]],<ref> See {{cite journal|last=Scholkopf|first=Bernhard|coauthors=Ralf Herbrich and 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|url=http://www.springerlink.com/content/v1tvba62hd4837h9/?MUD=MP}}</ref> <math>f(x_i) = \sum_{f=1}^n c_j \mathbf K_{ij}</math>, and <math> ||f||^2_{\mathcal H} = \langle f,f\rangle_\mathcal H = \sum_{i=1}^n\sum_{j=1}^n c_ic_jK(x_i,x_j) = c^T\mathbf K c </math>

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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(y_i,f(x_i)) = (1-yf(x))_+</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>{{cite journal|last=Lee|first=Yoonkyung|coauthors=Grace 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|url=http://www.tandfonline.com/doi/abs/10.1198/016214504000000098}}</ref> and with infinite data returns the [[Bayes' theorem|Bayes]] optimal solution:<ref>{{cite journal|last=Lin|first=Yi|title=Support Vector Machines and the Bayes Rule in Classification|journal=Data Mining and Knowledge Discovery|year=2002|month=July|volume=6|issue=3|pages=259–275|doi=10.1023/A:1015469627679|url=http://cbio.ensmp.fr/~jvert/svn/bibli/local/Lin2002Support.pdf}}</ref><ref>{{cite journal|last=Rosasco|first=Lorenzo|coauthors=Ernesto De Vito, Andrea Caponnetto, Michele Piana and Alessandro Verri|title=Are Loss Functions All the Same|journal=Neural Computation|year=2004|month=May|volume=16|series=5|pages=1063–1076|doi=10.1162/089976604773135104|url=http://www.mitpressjournals.org/doi/pdf/10.1162/089976604773135104}}</ref>

<math>f_b(x) = \left\{\begin{matrix}1&p(1|x)>p(-1|x)\\-1&p(1|x)<p(-1|x)\end{matrix}\right.</math>

==Derivation<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>==

To show that SVM is indeed a special case of Tikhonov regularization using the hinge loss, we will first state the Tikhonov regularization problem with the hinge loss, then demonstate that it is equivalent to traditional formulations of SVM. With the hinge loss, <math> V(y_i,f(x_i)) = (1-yf(x))_+</math> where <math>(s)_+ = max(s,0)</math>, the regularization problem becomes:

<math>f = \text{arg}\min_{f\in\mathcal{H}}\left\{\frac{1}{n}\sum_{i=1}^n (1-yf(x))_+ +\lambda||f||^2_\mathcal{H}\right\} </math>,

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This is equivalent to the standard SVM minimization problem.

==Notes and Referencesreferences==

{{Reflist}}