Regularization perspectives on support vector machines: Difference between revisions

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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]] 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">{{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 |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 |s2cid=261035640 |citeseerx=10.1.1.22.1879 }}</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 |author=Rosasco L. |author2=De Vito E. |author3=Caponnetto A. |author4=Piana M. |author5=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 |s2cid=11845688 }}</ref>

==Theoretical background==