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{{Infobox Algorithm
|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]]

Revision as of 17:04, 26 February 2013

Template:Infobox Algorithm Sequential minimal optimization (SMO) is an algorithm for solving the optimization problem which arises during the training of support vector machines. It was invented by John Platt in 1998 at Microsoft Research.[1] SMO is widely used for training support vector machines and is implemented by the popular LIBSVM tool.[2][3] 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.[4]

Optimization problem

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Consider a binary classification problem with a dataset (x1, y1), ..., (xn, yn), where xi is an input vector and Template:Nobr 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 form as follows:

subject to:

where C is an SVM hyperparameter and K(xi, xj) is the kernel function, both supplied by the user; and the variables are Lagrange multipliers.

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 , the smallest possible problem involves two such multipliers. Then, for any two multipliers and , the constraints are reduced to:

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

The algorithm proceeds as follows:

  1. Find a Lagrange multiplier that violates the Karush–Kuhn–Tucker (KKT) conditions for the optimization problem.
  2. Pick a second multiplier and optimize the pair .
  3. 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

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