Margin-infused relaxed algorithm

Margin-infused relaxed algorithm (MIRA) ^[1] is a machine learning and 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 a 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 ${\displaystyle T=\{x_{i},y_{i}\}}$Output: Set of parameters ${\displaystyle w}$

${\displaystyle i}$ ← 0, ${\displaystyle w^{(0)}}$ ← 0   for${\displaystyle n}$ ← 1 to${\displaystyle N}$for${\displaystyle t}$ ← 1 to${\displaystyle |T|}$${\displaystyle w^{(i+1)}}$ ← update ${\displaystyle w^{(i)}}$ according to ${\displaystyle \{x_{t},y_{t}\}}$${\displaystyle i}$ ← ${\displaystyle i+1}$end forend forreturn${\displaystyle {\frac {\sum _{j=1}^{N\times |T|}w^{(j)}}{N\times |T|}}}$

• "←" denotes assignment. For instance, "largest←item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the following value.

The update step is then formalized as a quadratic programming ^[2] problem: Find ${\displaystyle min\|w^{(i+1)}-w^{(i)}\|}$, so that ${\displaystyle score(x_{t},y_{t})-score(x_{t},y')\geq L(y_{t},y')\ \forall y'}$, i.e. the score of the current correct training ${\displaystyle y}$ must be greater than the score of any other possible ${\displaystyle y'}$ by at least the loss (number of errors) of that ${\displaystyle y'}$ in comparison to ${\displaystyle y}$.

References

1. Crammer, Koby; Singer, Yoram (2003). "Ultraconservative Online Algorithms for Multiclass Problems". Journal of Machine Learning Research . 3: 951–991.
2. McDonald, Ryan; Crammer, Koby; Pereira, Fernando (2005). "Online Large-Margin Training of Dependency Parsers" (PDF). Proceedings of the 43rd Annual Meeting of the ACL. Association for Computational Linguistics. pp. 91–98.
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.

External links

• adMIRAble – MIRA implementation in C++
• Miralium – MIRA implementation in Java
• MIRA implementation for Mahout in Hadoop