Papyrus 88 - Revision history

en>Trappist the monk: Fix CS1 deprecated coauthor parameter errors; using AWB

2014-07-21T12:33:37Z

Fix CS1 deprecated coauthor parameter errors; using AWB

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	~~\|image=~~
	~~\|class=[[Optimization algorithm]]~~ for ~~training support vector machines~~
	~~\|data=~~
	~~\|time=O(''n''³)~~
	~~\|space=~~
	}}
	~~'''Sequential minimal optimization''' ('''SMO''') is an algorithm for solving the optimization problem which arises during the training~~ of ~~[[support vector machine]]s~~. ~~It was invented by [[John Platt (Principal Researcher)\|John Platt]]~~ in ~~1998 at [[Microsoft Research]].<ref>{{Citation~~
	~~\| last = Platt \| first = John~~
	~~\| year = 1998~~
	~~\| title = Sequential Minimal Optimization: A Fast Algorithm~~ for ~~Training Support Vector Machines~~
	~~\| id = {{citeseerx\|10~~.~~1.1.43.4376}}~~
	~~}}</ref> SMO is widely used for training support vector machines~~ and ~~is implemented by~~ the ~~popular [[LIBSVM]] tool~~.~~<ref>{{cite journal~~
	~~\|last1=Chang \|first1=Chih-Chung~~
	~~\|last2=Lin \|first2=Chih-Jen~~
	~~\|title=LIBSVM: A library for support vector machines~~
	~~\|journal=ACM Transactions on Intelligent Systems~~ and ~~Technology~~
	~~\|volume=2 \|issue=3 \|year=2011~~
	~~}}</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 [~~[Quadratic programming\|QP]] solvers.<ref>{{Citation~~
	~~\| last = Rifkin \| first = Ryan~~
	~~\| year = 2002~~
	~~\| url =~~ http://~~dspace~~.~~mit~~.~~edu~~/~~handle~~/~~1721.1~~/~~17549~~
	~~\| title = Everything Old is New Again: a Fresh Look at Historical Approaches in Machine Learning~~
	~~\| journal = Ph.D. thesis~~
	~~\| pages = 18~~
	~~}}</ref>~~

	~~== Optimization problem ==~~
	~~{{main\|Support vector machine}}~~
	~~Consider a [[binary classification~~]~~] problem with a dataset (''x''<sub>1</sub>~~, ''y'~~'<sub>1</sub>)~~, ..~~., (''x''~~<~~sub~~>~~''n''~~<~~/sub~~>~~, ''y''~~<~~sub~~>~~''n''~~<~~/sub~~>~~), where ''x''<sub>''i''</sub> is~~ an ~~input vector and {{nobr\|''y''<sub>''i''</sub> ∈ {-1, +1} }} is a binary label corresponding~~ to it. ~~A soft-margin [[support vector machine]] is trained by solving a quadratic programming problem~~, ~~which~~ is ~~expressed in~~ the ~~[[Dual problem\|dual form]] as follows:~~

	:<~~math~~>~~\max_{\alpha} \sum_{i=1}^n \alpha_i - \frac12 \sum_{i=1}^n \sum_{j=1}^n y_i y_j K(x_i, x_j) \alpha_i \alpha_j,~~<~~/math~~>
	~~:subject to:~~
	~~:<math>0 \leq \alpha_i \leq C~~, ~~\quad \mbox{ for } i=~~1, ~~2, \ldots, n,</math>~~
	~~:<math>\sum_{i=1}^n y_i \alpha_i = 0</math>~~

	~~where ''C''~~ is an ~~SVM hyperparameter~~ and '~~'K''(''x''<sub>''i''</sub>~~, ~~''x''<sub>''j''</sub>) is~~ the ~~[[kernel function]], both supplied~~ by the ~~user;~~ and ~~the variables~~ <~~math~~>~~\alpha_i~~<~~/math~~> ~~are [[Lagrange multiplier]]~~s.

	~~== Algorithm ==~~
	~~SMO~~ is ~~an iterative algorithm~~ for ~~solving the optimization problem described above~~. ~~SMO breaks this problem into~~ a ~~series of smallest possible sub~~-~~problems, which are then solved analytically~~. ~~Because of the linear equality constraint involving the Lagrange multipliers~~ <~~math~~>~~\alpha_i~~<~~/math~~>, the ~~smallest possible problem involves two such multipliers~~. ~~Then, for any two multipliers~~ <~~math~~>~~\alpha_1~~<~~/math~~> and ~~<math>\alpha_2</math>,~~ the ~~constraints are reduced to:~~

	~~:<math>0 \leq \alpha_1~~, ~~\alpha_2 \leq C,</math>~~
	~~:<math>y_1 \alpha_1 + y_2 \alpha_2 = k,</math>~~

	~~and this reduced problem can be solved analytically:~~ one ~~needs to find a minimum of a one-dimensional quadratic function~~. <~~math~~>k<~~/math~~> ~~is the sum over the rest~~ of ~~terms~~ in ~~the equality constraint~~, which ~~is fixed in each iteration~~.

	~~The algorithm proceeds as follows:~~

	~~# Find a Lagrange multiplier~~ <~~math~~>~~\alpha_1~~<~~/math~~> ~~that violates the [[Karush–Kuhn–Tucker conditions\|Karush–Kuhn–Tucker (KKT) conditions]] for the optimization problem~~.
	~~# Pick a second multiplier <math>\alpha_2</math>~~ and ~~optimize~~ the ~~pair <math>(\alpha_1,\alpha_2)</math>~~.
	~~# 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.~~

	~~== References ==~~
	~~{{reflist}}~~

	~~== External links ==~~
	* [http://www.csie.~~ntu.edu.tw/~cjlin/libsvm/ LIBSVM]~~

	~~{{DEFAULTSORT:Sequential Minimal Optimization}}~~
	~~[[Category:Optimization algorithms and methods]]~~
	~~[[Category:Support vector machines]]~~

en>Leszek Jańczuk: /* Further reading */

2013-02-26T15:04:39Z

en>Leszek Jańczuk: ref

2011-08-27T17:14:22Z

ref

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