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'''Margin Infused Relaxed Algorithm (MIRA)'''<ref name="crammer-singer2003">
{{cite journal
| last1 = Crammer | first1 = Koby
| last2 = Singer | first2 = Yoram
| year = 2003
| title = Ultraconservative Online Algorithms for Multiclass Problems
| journal = [[Journal of Machine Learning Research]]
| volume = 3
| pages = 951–991
| url = http://jmlr.csail.mit.edu/papers/v3/crammer03a.html
}}
</ref> is a [[machine learning]] algorithm, an [[online algorithm]] for [[multiclass classification]] problems. It is designed to learn a set of parameters (vector or matrix) by processing all the given training examples one-by-one and updating the parameters according to each training example, so that the current training example is classified correctly with a [[margin (machine learning)|margin]] against incorrect classifications at least as large as their loss.<ref name="mcdonald-etal2005">
{{cite conference
| last1 = McDonald | first1 = Ryan
| last2 = Crammer | first2 = Koby
| last3 = Pereira | first3 = Fernando
| title = Online Large-Margin Training of Dependency Parsers
| booktitle = Proceedings of the 43rd Annual Meeting of the ACL
| publisher = [[Association for Computational Linguistics]]
| date = 2005
| pages = 91–98
| url = http://aclweb.org/anthology-new/P/P05/P05-1012.pdf }}
</ref> The change of the parameters is kept as small as possible.


A two-class version called '''binary MIRA'''<ref name="crammer-singer2003"/> simplifies the algorithm by not requiring the solution of a [[quadratic programming]] problem (see below). When used in an [[one-vs.-all]] configuration, binary MIRA can be extended to a multiclass learner that approximates full MIRA, but may be faster to train.


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The flow of the algorithm<ref>Watanabe, T. et al (2007): ''Online Large Margin Training for Statistical Machine Translation''. In: Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational
Natural Language Learning, 764–773.</ref><ref>Bohnet, B. (2009): ''Efficient Parsing of Syntactic and Semantic Dependency Structures''. Proceedings of Conference on Natural Language Learning (CoNLL), Boulder, 67-72.</ref> looks as follows:
 
{{Algorithm-begin|name=MIRA}}
  '''Input:''' Training examples <math>T = \{x_i, y_i\}</math>
  '''Output:''' Set of parameters <math>w</math>
 
  <math>i</math> ← 0, <math>w^{(0)}</math> ← 0
  '''for''' <math>n</math> ← 1 '''to''' <math>N</math>
    '''for''' <math>t</math> ← 1 '''to''' <math>|T|</math>
      <math>w^{(i+1)}</math> ← update <math>w^{(i)}</math> according to <math>\{x_t, y_t\}</math>
      <math>i</math> ← <math>i + 1</math>
    '''end for'''
  '''end for'''
  '''return''' <math>\frac{\sum_{j=1}^{N \times |T|} w^{(j)}}{N \times |T|}</math>
{{Algorithm-end}}
 
The update step is then formalized as a [[quadratic programming]]<ref name="mcdonald-etal2005"/> problem: Find <math>min\|w^{(i+1)} - w^{(i)}\|</math>, so that <math>score(x_t,y_t) - score(x_t,y')\geq L(y_t,y')\ \forall y'</math>, i.e. the score of the current correct training <math>y</math> must be greater than the score of any other possible <math>y'</math> by at least the loss (number of errors) of that <math>y'</math> in comparison to <math>y</math>.
 
==References==
<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist}}
 
==External links==
*[https://github.com/jihelhere/adMIRAble adMIRAble] - MIRA implementation in C++
*[https://code.google.com/p/miralium/ Miralium] - MIRA implementation in Java
*[https://cwiki.apache.org/confluence/display/MAHOUT/Online+Passive+Aggressive MIRA implementation] for [[Apache Mahout|Mahout]] in [[Hadoop]]
 
[[Category:Classification algorithms]]

Latest revision as of 07:35, 19 October 2013

Margin Infused Relaxed Algorithm (MIRA)[1] is a machine learning algorithm, an online algorithm for multiclass classification problems. It is designed to learn a set of parameters (vector or matrix) by processing all the given training examples one-by-one and updating the parameters according to each training example, so that the current training example is classified correctly with a margin against incorrect classifications at least as large as their loss.[2] The change of the parameters is kept as small as possible.

A two-class version called binary MIRA[1] simplifies the algorithm by not requiring the solution of a quadratic programming problem (see below). When used in an one-vs.-all configuration, binary MIRA can be extended to a multiclass learner that approximates full MIRA, but may be faster to train.

The flow of the algorithm[3][4] looks as follows:

Template:Algorithm-begin 

  Input: Training examples T={xi,yi}
  Output: Set of parameters w

  i ← 0, w(0) ← 0
  for n ← 1 to N
    for t ← 1 to |T|
      w(i+1) ← update w(i) according to {xt,yt}
      i ← i+1
    end for
  end for
  return ∑j=1N×|T|w(j)N×|T|

Template:Algorithm-end

The update step is then formalized as a quadratic programming[2] problem: Find min‖w(i+1)−w(i)‖, so that score(xt,yt)−score(xt,y′)≥L(yt,y′)∀y′, i.e. the score of the current correct training y must be greater than the score of any other possible y′ by at least the loss (number of errors) of that y′ in comparison to y.

References

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  3. ↑ Watanabe, T. et al (2007): Online Large Margin Training for Statistical Machine Translation. In: Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, 764–773.
  4. ↑ Bohnet, B. (2009): Efficient Parsing of Syntactic and Semantic Dependency Structures. Proceedings of Conference on Natural Language Learning (CoNLL), Boulder, 67-72.
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