On the Existence of Linear Weak Learners and Applications to Boosting
We consider the existence of a linear weak learner for boosting algorithms. A weak learner for binary classification problems is required to achieve a weighted empirical error on the training set which is bounded from above by 1/2 − γ, γ > 0, for any distribution on the data set. Moreover, in order that the weak learner be useful in terms of generalization, γ must be sufficiently far from zero. While the existence of weak learners is essential to the success of boosting algorithms, a proof of their existence based on a geometric point of view has been hitherto lacking. In this work we show that under certain natural conditions on the data set, a linear classifier is indeed a weak learner. Our results can be directly applied to generalization error bounds for boosting, leading to closed-form bounds. We also provide a procedure for dynamically determining the number of boosting iterations required to achieve low generalization error. The bounds established in this work are based on the theory of geometric discrepancy.