Gimpel and Smith, NAACL 2010

Softmax-Margin CRFs: Training Log-Linear Models with Cost Functions

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Citation

Kevin Gimpel and Noah A. Smith. Softmax-margin CRFs: Training log-linear models with loss functions. In Proceedings of the Human Language Technologies Conference of the North American Chapter of the Association for Computational Linguistics, pages 733-736, Los Angeles, California, USA, June 2010.

Summary

The authors want to be able to incorporate a cost function (present in structured SVMs) into standard conditional log-likelihood models. They introduce the softmax-margin objective function that achieves the best of both worlds. Using a NER task, it performs significantly better than a standard conditional loglikelihood model, a max-margin model, and the perceptron, but is indistinguishable from MIRA, risk, and JRB (Jensen risk bound; defined in the paper).

Brief Description of the Softmax-Margin objective function

Consider the objective functions for these four methods. Our

Conditional log likelihood: $\min _{\theta }\sum _{i=1}^{n}-{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y^{(i)})+\log \sum _{y\in {\mathcal {Y}}(x^{(i)})}\exp\{{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y)\}$

Max-margin: $\min _{\theta }\sum _{i=1}^{n}-{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y^{(i)})+\max _{y\in {\mathcal {Y}}(x^{(i)})}({\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y)+cost(y^{(i)},y))$

Risk: $\min _{\theta }\sum _{i=1}^{n}\sum _{y\in {\mathcal {Y}}(x^{(i)})}cost(y^{(i)},y){\dfrac {\exp\{{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y)\}}{\sum _{y'\in {\mathcal {Y}}(x^{(i)})}\exp\{{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y')\}}}$

Softmax-margin: $\min _{\theta }\sum _{i=1}^{n}-{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y^{(i)})+\log \sum _{y\in {\mathcal {Y}}(x^{(i)})}\exp\{{\boldsymbol {\theta }}^{T}{\boldsymbol {f}}(x^{(i)},y)+cost(y^{(i)},y)\}$