Gradient Boosting: Fostering accuracy even further

As in many real-world situations, union makes algorithms stronger. With this philosophy in mind, ensemble methods combine several weak classifiers into a massive one---in terms of accuracy. In the last post we learnt a primer with Random Forest. Therefore, the next cornerstone is gradient boosting. I mentioned Gradient Boosting many times in this blog, but I only commented the fundamental ideas, without discussing further the details. In this entry I will share my two cents. Let me introduce a little bit of history, first: recall the Kaggle-Higgs competition. The top scores in the leaderboard have been obtained by using distinct forms of gradient boosting, and XGBoost is the direct responsible of many of these. The question is, hence, how does this algorithm work?

Figure 1. A high-level description of the Gradient Boosting method I programmed. Click to enlarge.

Informally, Gradient Boosting generates a sequence of classifiers in the form of an additive expansion, that is, at each iteration a new weak classifier is trained to improve the accuracy of the previous ones, in the same fashion as AdaBoost. However, and differently from this one, the gradient of the loss function (i.e., the function used to check how well our algorithm performs) is used to guide the search toward a very accurate solution (i.e., continual improvement by means of a gradient). Another characteristic is that Gradient Boosting works best with a regression tree as a base model. Combining several of those, the common loss function used is the mean squared error, which is differentiable (and also straightforward to handle). Figure 1 shows the high-level description of the algorithm.

So far, so good, we have an algorithm that is straightforward to implement and that offers a high degree of accuracy. However, the trick lies in the trees. Gradient Boosting exploits the benefits of well-inducted trees, and this is, probably, the most difficult part of the technique. There are many possibilities, and in my particular case I implemented the simplest form of a binary regression tree as a basis. This tree performs the splits by computing the gain in variance (informally, how much variety is in the output variable), and without considering any form of backfitting (i.e., tree pruning). Also, I just “ignored” the unknown values, making the variance computation with the known available. Despite this rather simplistic approach, I obtained a score of 3.31 in the Approximate Median Significance (AMS) metric (almost the same as in the original Higgs challenge reference study, which is of 3.34!). The configuration was pretty out-of-the-shelf, consisting of 300 boosted trees, a learning rate of 0.1 and a limitation of 10 leaves per tree. Just for instance, XGBoost uses unpruned binary trees, and its amazing accuracy comes from the tricks used for growing the trees (e.g., mainly handling unknown values).

As concluding remarks, I have to say that I enjoyed the experience. OK, I did not win the challenge (the leader has a score of 3.85 in the aforementioned AMS metric) but I learned a lot, discovering---and, of course, implementing---these marvelous machine learning techniques. Following the rules of the challenge, I will upload and share the source code (it is programmed in a user-friendly C++) when it finishes (September the 15th).

References

[1] G. James, D. Witten, T. Hastie and R. Tiibshirani, "An Introduction to Statistical Learning With Applications in R," ISSN 1431-875X. Springer, 2013.

[2] J. Friedman, "Greedy function approximation: Gradient boosting machine," the Annals of Statistics 2001, Vol. 29, No. 5, 1189–1232.