Sequential minimal optimization: Difference between revisions

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{{short description|Algorithm for solving the quadratic programming problem from training SVMs}}

{{Infobox Algorithm

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'''Sequential minimal optimization''' ('''SMO''') is an algorithm for solving the [[quadratic programming]] (QP) problem that arises during the training of [[support -vector machine]]s (SVM). It was invented by [[John Platt (Principalcomputer Researcherscientist)|John Platt]] in 1998 at [[Microsoft Research]].<ref name = "Platt">{{CitationCite web

| last = Platt | first = John

| year = 1998

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|journal=ACM Transactions on Intelligent Systems and Technology

|volume=2 |issue=3 |year=2011

|doi=10.1145/1961189.1961199

}}</ref><ref>Luca Zanni (2006). ''[http://jmlr.csail.mit.edu/papers/volume7/zanni06a/zanni06a.pdf Parallel Software for Training Large Scale Support Vector Machines on Multiprocessor Systems]''.</ref> The publication of the SMO algorithm in 1998 has generated a lot of excitement in the SVM community, as previously available methods for SVM training were much more complex and required expensive third-party QP solvers.<ref>{{Citation

|s2cid=961425

}}</ref><ref>{{cite web |first=Luca |last=Zanni (|date=2006). ''[|url=http://jmlr.csail.mit.edu/papers/volume7/zanni06a/zanni06a.pdf |title=Parallel Software for Training Large Scale Support Vector Machines on Multiprocessor Systems]''. }}</ref> The publication of the SMO algorithm in 1998 has generated a lot of excitement in the SVM community, as previously available methods for SVM training were much more complex and required expensive third-party QP solvers.<ref>{{Citationcite thesis

| last = Rifkin | first = Ryan

| year = 2002

| url hdl= http://dspace.mit.edu/handle/1721.1/17549

| title = Everything Old is New Again: a Fresh Look at Historical Approaches in Machine Learning

| type = Ph.D. Thesis |publisher=Massachusetts Institute of Technology

| journal = Ph.D. thesis

| pages = 18

}}</ref>

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# Repeat steps 1 and 2 until convergence.

When all the Lagrange multipliers satisfy the KKT conditions (within a user-defined tolerance), the problem has been solved. Although this algorithm is guaranteed to converge, heuristics are used to choose the pair of multipliers so as to accelerate the rate of convergence. This is critical for large data sets since there are <math>n(n − -1)/2</math> possible choices for <math>\alpha_i</math> and <math>\alpha_j</math>.

== Related Workwork ==

The first approach to splitting large SVM learning problems into a series of smaller optimization tasks was proposed by [[Bernhard E Boser]], [[Isabelle M Guyon]], and [[Vladimir Vapnik]].<ref name="ReferenceA">{{Cite book | doi = 10.1145/130385.130401| chapter = A training algorithm for optimal margin classifiers| title = Proceedings of the fifth annual workshop on Computational learning theory - COLT '92| pages = 144| year = 1992| last1 = Boser | first1 = B. E. | last2 = Guyon | first2 = I. M. | last3 = Vapnik | first3 = V. N. | isbn = 089791497X978-0897914970| citeseerx = 10.1.1.21.3818| s2cid = 207165665}}</ref> It is known as the "chunking algorithm". The algorithm starts with a random subset of the data, solves this problem, and iteratively adds examples which violate the optimality conditions. One disadvantage of this algorithm is that it is necessary to solve QP-problems scaling with the number of SVs. On real world sparse data sets, SMO can be more than 1000 times faster than the chunking algorithm.<ref name = "Platt"/>

In 1997, [[E. Osuna]], [[R. Freund]], and [[F. Girosi]] proved a theorem which suggests a whole new set of QP algorithms for SVMs.<ref>{{Cite book | doi = 10.1109/NNSP.1997.622408| chapter = An improved training algorithm for support vector machines| title = Neural Networks for Signal Processing [1997] VII. Proceedings of the 1997 IEEE Workshop| pages = 276 - 285276–285| year = 1997| last1 = Osuna | first1 = E. | last2 = Freund | first2 = R. | last3 = Girosi | first3 = F. | isbn = 978-0-7803-4256-9| citeseerx = 10.1.1.392.7405| s2cid = 5667586}}</ref> By the virtue of this theorem a large QP problem can be broken down into a series of smaller QP sub-problems. A sequence of QP sub-problems that always add at least one violator of the [[Karush–Kuhn–Tucker conditions|Karush–Kuhn–Tucker (KKT) conditions]] is guaranteed to converge. The chunking algorithm obeys the conditions of the theorem, and hence will converge.<ref name = "Platt"/> The SMO algorithm can be considered a special case of the Osuna algorithm, where the size of the optimization is two and both Lagrange multipliers are replaced at every step with new multipliers that are chosen via good heuristics.<ref name = "Platt"/>

The SMO algorithm is closely related to a family of optimization algorithms called [[Bregman method]]s or row-action methods. These methods solve convex programming problems with linear constraints. They are iterative methods where each step projects the current primal point onto each constraint.<ref name = "Platt"/>

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== References ==

{{reflist|30em}}

{{Optimization algorithms}}

[[Category:Optimization algorithms and methods]]

[[Category:Support vector machines]]