Ando and Zhang ACL 2005

Under Construction

A high-performance semi-supervised learning method for text chunking, by R. K. Ando, T. Zhang. In Proceedings of the 43rd Annual Meeting on Association for Computational Linguistics, 2005.

This Paper is available online [1].

Summary

This paper describes an exponentiated gradient (EG) algorithm for training conditional log-linear models. Conditional log-linear models are used for several key structured prediction tasks such as NER, POS tagging, Parsing.

In this paper, they propose a fast and efficient algorithm for optimizing log-linear models such as CRFs.

The common practice of optimizing the conditional log likelihood of a CRF is often via conjugate-gradient or L-BFGS algorithms (Sha & Pereira, 2003), which typically would require at least one pass through the entire dataset before updating the weight vector. The author's approach here is an online algorithm based on exponentiated gradient updates (Kevin & Warmuth, 1997).

Brief description of the method

Consider a supervised learning setting with objects $x\in {\mathcal {X}}$ and corresponding labels $y\in {\mathcal {Y}}$, which maybe trees, sequences or other high dimensional structure. Also, assume we are given a function $\phi (x,y)$ that maps $(x,y)$ pairs to feature vectors ${\mathcal {R}}^{d}$. Given a parameter vector $\mathbf {w} \in {\mathcal {R}}^{d}$, a conditional log-linear model defines a distribution over labels as:

$p(y|x;\mathbf {w} )={\frac {1}{Z_{x}}}\exp \left(\mathbf {w} \phi (x,y)\right)$

where $Z_{x}$ is a partition function.

The problem of learning $\mathbf {w}$ from the training data is thus finding $\mathbf {w}$ which maximizes the regularized log-likelihood:

$\mathbf {w} ^{*}=\arg \max _{w}\sum _{i}\log p(y_{i}|x_{i};\mathbf {w} )-{\frac {C}{2}}\lVert \mathbf {w} \rVert ^{2}$

where $C$ is the regularization parameter. The above equation has a convex dual which is derived in Lebanon and Lafferty NIPS 2001. With dual variables $\alpha _{i,y}$, and $\mathbf {\alpha } =[\mathbf {\alpha } _{1},\mathbf {\alpha } _{2},\cdots ,\mathbf {\alpha } _{n}]$, we define:

$Q(\mathbf {\alpha } )=\sum _{i}\sum _{y}\alpha _{i,y}\log \alpha _{i,y}+{\frac {1}{2C}}\lVert \mathbf {w} (\alpha )\rVert ^{2}$

where $\mathbf {w} (\alpha )=\sum _{i}\sum _{y}\alpha _{i,y}\left(\phi (x_{i},y_{i})-\phi (x_{i},y)\right)$

The dual problem is thus

$\alpha ^{*}=\arg \min _{\alpha \in \Delta ^{n}}Q(\alpha )$

EG Algorithm

Given a set of distributions $\alpha \in \Delta ^{n}$, the update equations are

$\alpha _{i,y}^{'}={\frac {1}{Z}}\alpha _{i,y}\exp(-\eta \nabla _{i,y})$

where

$Z_{i}=\sum _{\hat {y}}\alpha _{i,{\hat {y}}}\exp(-\eta \nabla _{i,{\hat {y}}})$

and

$\nabla {i,y}={\frac {\partial Q(\alpha )}{\partial \alpha _{i,y}}}=1+\log \alpha _{i,y}+{\frac {1}{C}}\mathbf {w} (\alpha )\cdot \left(\phi (x_{i},y_{i})-\phi (x_{i},y)\right)$

Batch learning

At each iteration, $\alpha '$ is updated simultaneously with all (or subset of) the available training instances.

Online learning

At each iteration, we choose a single training instance, and update $\alpha '$

Convergence rate of batch algorithm

To get within $\epsilon$ of the optimum parameters, we need $O({\frac {1}{\eta \epsilon }})$ iterations.

Experimental Result

The authors compared the performance of the EG algorithm to conjugated-gradient and L-BFGS methods.

Multiclass classification

The authors used a subset of the MNIST handwritten digits classification.

It can be seen that the EG algorithm converges considerably faster than the other methods.

Structured learning (dependency parsing)

The author used the Slovene data in UsesDataset:CoNLL-X Shared Task on Multilingual dependency parsing.

It can be seen that the EG algorithm converges faster in terms of objective function and accuracy measures.

Related Papers

The approach here is also similar to the use of EG algorithms for large margin structured classification in Bartlett et al NIPS 2004.