Margin Infused Relaxed Algorithm

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:

Algorithm MIRA
  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 $\frac{\sum_{j=1}^{N \times |T|} w^{(j)}}{N \times |T|}$

• "←" is a loose shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the value that follows.

The update step is then formalized as a quadratic programming^[2] problem: Find $min\|w^{(i+1)} - w^{(i)}\|$, so that $score(x_t,y_t) - score(x_t,y')\geq L(y_t,y')\ \forall y'$, 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

1. ^ ^{a b} Crammer, K., Singer, Y. (2003): Ultraconservative Online Algorithms for Multiclass Problems. In: Journal of Machine Learning Research 3, 951-991. http://jmlr.csail.mit.edu/papers/v3/crammer03a.html
2. ^ ^{a b} McDonald, R., K. Crammer and F.C.N. Pereira (2005): Online Large-Margin Training of Dependency Parsers. In: Proceedings of the 43rd Annual Meeting of the ACL, pp. 91-98. http://aclweb.org/anthology-new/P/P05/P05-1012.pdf
3. ^ Wanatabe, 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.

External links