Numerical data poses a problem to symbolic learning methods, since numerical value ranges inherently need to be partitioned into intervals for representation and handling. An evaluation function is used to approximate the goodness of different partition candidates. Most existing methods for multisplitting on numerical attributes axe based on heuristics, because of the apparent efficiency advantages. We characterize a class of well-behaved cumulative evaluation functions for which efficient discovery of the optimal multisplit is possible by dynamic programming. A single pass through the data suffices to evaluate multisplits of all axities. This class contains many important attribute evaluation functions familiar from symbolic machine learning research. Our empirical experiments convey that there is no significant differences in efficiency between the method that produces optimM partitions and those that are based on heuristics. Moreover~ we demonstrate that optimal multisplitting can be beneficial in decision tree learning in contrast to using the much applied binarization of numerical attributes or heuristical multisplitting.

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 consider multisplitting of numerical value ranges, a task that is encountered as a discretization step preceding induction and also embedded into learning algorithms. We are interested in finding the partition that optimizes the value of a given attribute evaluation function. For most commonly used evaluation functions this task takes quadratic time in the number of potential cut points in the numerical range. Hence, it is a potential bottleneck in data mining algorithms.We present two techniques that speed up the optimal multisplitting task. The first one aims at discarding cut point candidates in a quick linear-time preprocessing scan before embarking on the actual search. We generalize the definition of boundary points by Fayyad and Irani to allow us to merge adjacent example blocks that have the same relative class distribution. We prove for several commonly used evaluation functions that this processing removes only suboptimal cut points. Hence, the algorithm does not lose optimality.Our second technique tackles the quadratic-time dynamic programming algorithm, which is the best schema for optimizing many well-known evaluation functions. We present a technique that dynamically-i.e., during the search-prunes partitions of prefixes of the sorted data from the search space of the algorithm. The method works for all convex and cumulative evaluation functions.Together the use of these two techniques speeds up the multisplitting process considerably. Compared to the baseline dynamic programming algorithm the speed-up is around 50 percent on the average and up to 90 percent in some cases. We conclude that optimal multisplitting is fully feasible on all benchmark data sets we have encountered.

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