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Top-down induction of decision trees has been observed to suffer from the inadequate functioning of the pruning phase. In particular, it is known that the size of the resulting tree grows linearly with the sample size, even though the accuracy of the tree does not improve. Reduced Error Pruning is an algorithm that has been used as a representative technique in attempts to explain the problems of decision tree learning. In this paper we present analyses of Reduced Error Pruning in three different settings. First we study the basic algorithmic properties of the method, properties that hold independent of the input decision tree and pruning examples. Then we examine a situation that intuitively should lead to the subtree under consideration to be replaced by a leaf node, one in which the class label and attribute values of the pruning examples are independent of each other. This analysis is conducted under two different assumptions. The general analysis shows that the pruning probability of a node fitting pure noise is bounded by a function that decreases exponentially as the size of the tree grows. In a specific analysis we assume that the examples are distributed uniformly to the tree. This assumption lets us approximate the number of subtrees that are pruned because they do not receive any pruning examples. This paper clarifies the different variants of the Reduced Error Pruning algorithm, brings new insight to its algorithmic properties, analyses the algorithm with less imposed assumptions than before, and includes the previously overlooked empty subtrees to the analysis

Abstract. We present two new methods for obtaining generalization error bounds in a semi-supervised setting. Both methods are based on approximating the disagreement probability of pairs of classifiers using unlabeled data. The first method works in the realizable case. It suggests how the ERM principle can be refined using unlabeled data and has provable optimality guarantees when the number of unlabeled examples is large. Furthermore, the technique extends easily to cover active learning. A downside is that the method is of little use in practice due to its limitation to the realizable case.The idea in our second method is to use unlabeled data to transform bounds for randomized classifiers into bounds for simpler deterministic classifiers. As a concrete example of how the general method works in practice, we apply it to a bound based on cross-validation. The result is a semi-supervised bound for classifiers learned based on all the labeled data. The bound is easy to implement and apply and should be tight whenever cross-validation makes sense. Applying the bound to SVMs on the MNIST benchmark data set gives results that suggest that the bound may be tight enough to be useful in practice.

We compare the practical performance of several recently proposed algorithms for active learning in the online classification setting. We consider two active learning algorithms (and their combined variants) that are strongly online, in that they access the data sequentially and do not store any previously labeled examples, and for which formal guarantees have recently been proven under various assumptions. We motivate an optical character recognition (OCR) application that we argue to be appropriately served by online active learning. We compare the practical efficacy, for this application, of the algorithm variants, and show significant reductions in label-complexity over random sampling.

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