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 minimizes 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=https://www.mit.edu/~9.520/spring12/slides/class06/class06_RLSSVM.pdf}}</ref>

Regularization perspectives on support-vector machines interpret SVM as a special case of 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 |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]], [[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}}</ref><ref name="Lee 2012 67–81">{{cite journal |lastlast1=Lee |firstfirst1=Yoonkyung |author1-link= Yoonkyung Lee |first2=Grace |last2=Wahba |author2-link=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 }}</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==

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* {{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&pg=PR7&dqq=vapnik+the+nature+of+statistical+learning+theory#v=onepage&qpg=vapnik%20the%20nature%20of%20statistical%20learning%20theory&f=falsePR7}}

[[Category:Support vector machines]]