CoBoost is a semi-supervised training algorithm proposed by Collins and Singer in 1999. The original application for the algorithm was the task of named-entity recognition using very weak learners, but it can be used for performing semi-supervised learning in cases where data features may be redundant.

It may be seen as a combination of co-training and boosting. Each example is available in two views (subsections of the feature set), and boosting is applied iteratively in alternation with each view using predicted labels produced in the alternate view on the previous iteration. CoBoosting is not a valid boosting algorithm in the PAC learning sense.

Motivation

CoBoosting was an attempt by Collins and Singer to improve on previous attempts to leverage redundancy in features for training classifiers in a semi-supervised fashion. CoTraining, a seminal work by Blum and Mitchell, was shown to be a powerful framework for learning classifiers given a small number of seed examples by iteratively inducing rules in a decision list. The advantage of CoBoosting to CoTraining is that it generalizes the CoTraining pattern so that it could be used with any classifier. CoBoosting accomplishes this feat by borrowing concepts from AdaBoost.

. While ideal, this constraint is in fact too strong due to noise and other factors, and both algorithms instead seek to maximize the agreement between the two functions. The second property is that the two views must not be highly correlated.

Algorithm

.

:

Set pseudo-labels:

that minimizes expanded training error.

that minimizes expanded training error.

Update the value for current strong non-thresholded classifier:

The final strong classifier output is

Setting up AdaBoost

CoBoosting builds on the AdaBoost algorithm, which gives CoBoosting its generalization ability since AdaBoost can be used in conjunction with many other learning algorithms. This build up assumes a two class classification task, although it can be adapted to multiple class classification. In the AdaBoost framework, weak classifiers are generated in series as well as a distribution over examples in the training set. Each weak classifier is given a weight and the final strong classifier is defined as the sign of the sum of the weak classifiers weighted by their assigned weight. (See AdaBoost Wikipedia page for notation). In the AdaBoost framework Schapire and Singer have shown that the training error is bounded by the following equation:

we get:

is the feature selected in the current weak hypothesis. Three equations are defined describing the sum of the distributions for in which the current hypothesis has selected either correct or incorrect label. Note that it is possible for the classifier to abstain from selecting a label for an example, in which the label provided is 0. The two labels are selected to be either -1 or 1.

to be as follows:

from this equation we get:

The training error thus is minimized by selecting the weak hypothesis at every iteration that minimizes the previous equation.

AdaBoost with two views

. The algorithm trains two classifiers in the same fashion as AdaBoost:

iteration we can set the pseudo-labels for the jth update to be:

. Where

is defined as follows:

can be rewritten as

is minimized in a greedy fashion.

References

Footnotes