Papyrus 88: Difference between revisions

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{{Infobox Algorithm
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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]]

Latest revision as of 13:33, 21 July 2014

Gold has for long been a signifier of constant and efficient money. Individuals have been purchasing and investing in gold for hundreds of years. Even these days, gold continues to be the most secure and most profitable financial commitment chance of those around the world. However, today's gold rate has made people careful and put gold out of some people's affordability. Indian could possibly be the http://trading.goldgrey.org/category/futures-trading/; just click the following internet page, biggest gold customer and it's that isn't going to change any time soon. Traders in India don't see Gold as suspect, just how many American investors do. This is due to the fact of India's long connections to Gold that expand back thousands of years.



You can also investigate forms of payment as an incentive for people to pay you promptly. Do you offer bank transfer or credit card facilities or bill payment facilities? The more options people have of paying, the easier it is for them to action the payment.

As numerous know, you can find incredibly intelligent folks on both sides of the gold trade. Both sides are incredibly passionate in their beliefs that gold will possibly drastically go up or even down. 1 thing in which appears evident, is the fact that gold is no longer exclusively employed as an inflationary hedge. Factors like QE3 will constantly influence the buying price of gold and will elevate the rising tide of gold significant caps, ETF's and gold small cap stocks. Nevertheless, a lot of the move we have observed within this precious metal could be caused by probably the most fundamental economic principal. It's known as supply and demand.

I know all to nicely about staying broke and not having gold in Earth Of Warcraft to the point that it get's irritating. This gold manual is then for you penny-significantly less and annoyed. As you know by now in Entire world Of Warcraft you use gold to degree up your characters and to acquire all types of things. The Universe Of Warcraft gold market is as a result a quite intricate and lower-throat a person. Illegal gold trading in World Of Warcraft is believed at around $1billion per yr.

If you choose to sell your items online, you should be extra careful. Pick a legitimate buyer by contacting the person to make sure that he or she is serious to buy your gold pieces.

Forex trading industry would be the largest and also the fastest increasing industry on earth. Its every day turnover is more than 2.5 trillion dollars, that's one hundred times increased than the NASDAQ everyday turnover.

There are numerous varied strategies of creating heaps of gold in Globe Of Warcraft and absolutely everyone has their unique way or favourite way. Some gold earning techniques will make gold faster for you than other individuals , so consider them all and see which works for you most effective.

Investing in valuable metals is simple. Right knowledge and understanding about the gold trade will surely help. Also equipping yourself along with the latest facts on the gold market will help you in making transactions and dealings. Being sharp and watchful to the daily fluctuation of gold in the global market are keys to sell bullion at best amount.

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