Boosted tree classifier derived from cv.DTrees

Boosting

A common machine learning task is supervised learning. In supervised learning, the goal is to learn the functional relationship F: y = F(x) between the input x and the output y. Predicting the qualitative output is called classification, while predicting the quantitative output is called regression.

Boosting is a powerful learning concept that provides a solution to the supervised classification learning task. It combines the performance of many weak classifiers to produce a powerful committee [HTF01]. A weak classifier is only required to be better than chance, and thus can be very simple and computationally inexpensive. However, many of them smartly combine results to a strong classifier that often outperforms most monolithic strong classifiers such as SVMs and Neural Networks.

Decision trees are the most popular weak classifiers used in boosting schemes. Often the simplest decision trees with only a single split node per tree (called stumps) are sufficient.

The boosted model is based on N training examples (xi,yi), i=1..N with xi IN R^k and yi IN {-1,+1}. xi is a K-component vector. Each component encodes a feature relevant to the learning task at hand. The desired two-class output is encoded as -1 and +1.

Different variants of boosting are known as Discrete Adaboost, Real AdaBoost, LogitBoost, and Gentle AdaBoost [FHT98]. All of them are very similar in their overall structure. Therefore, this chapter focuses only on the standard two-class Discrete AdaBoost algorithm, outlined below. Initially the same weight is assigned to each sample (step 2). Then, a weak classifier f_m(x) is trained on the weighted training data (step 3a). Its weighted training error and scaling factor c_m is computed (step 3b). The weights are increased for training samples that have been misclassified (step 3c). All weights are then normalized, and the process of finding the next weak classifier continues for another M-1 times. The final classifier F(x) is the sign of the weighted sum over the individual weak classifiers (step 4).

Two-class Discrete AdaBoost Algorithm

1. Set N examples (xi,yi), i=1..N with xi IN R^K, yi IN {-1,+1}.
2. Assign weights as wi = 1/N, i=1..N.
3. Repeat for m=1,2,...,M:
   1. Fit the classifier f_m(x) IN {-1,1}, using weights wi on the training data.
   2. Compute err_m = Ew[1_(y!=f_m(x))], c_m = log((1-err_m)/err_m).
   3. Set wi = wi * exp[c_m * 1(yi!=f_m(xi))], i=1,2,...,N, and renormalize so that sum_i{wi}=1.
4. Classify new samples x using the formula: sign(sum{m} = 1 * M * c_m * f_m(x)).

Note: Similar to the classical boosting methods, the current implementation supports two-class classifiers only. For M > 2 classes, there is the AdaBoost.MH algorithm (described in [FHT98]) that reduces the problem to the two-class problem, yet with a much larger training set.

To reduce computation time for boosted models without substantially losing accuracy, the influence trimming technique can be employed. As the training algorithm proceeds and the number of trees in the ensemble is increased, a larger number of the training samples are classified correctly and with increasing confidence, thereby those samples receive smaller weights on the subsequent iterations. Examples with a very low relative weight have a small impact on the weak classifier training. Thus, such examples may be excluded during the weak classifier training without having much effect on the induced classifier. This process is controlled with the WeightTrimRate parameter. Only examples with the summary fraction WeightTrimRate of the total weight mass are used in the weak classifier training. Note that the weights for all training examples are recomputed at each training iteration. Examples deleted at a particular iteration may be used again for learning some of the weak classifiers further [FHT98].

Prediction with Boost

The cv.Boost.predict method should be used. Pass RawOutput=true to get the raw sum from Boost classifier.

References

[HTF01]:

Hastie Trevor, Tibshirani Robert, and Friedman Jerome. "The elements of statistical learning: data mining, inference and prediction". New York: Springer-Verlag, 1(8):371-406, 2001.

[FHT98]:

Jerome Friedman, Trevor Hastie, and Robert Tibshirani. "Additive Logistic Regression: a Statistical View of Boosting". Technical Report, Dept. of Statistics, Stanford University, 1998.