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We analyze the performance of the top-down multiclass classificationalgorithm for decision tree learning called LOMtree, recently proposed in theliterature Choromanska and Langford (2014) for solving efficientlyclassification problems with very large number of classes. The algorithm onlineoptimizes the objective function which simultaneously controls the depth of thetree and its statistical accuracy. We prove important properties of thisobjective and explore its connection to three well-known entropy-based decisiontree objectives, i.e. Shannon entropy, Gini-entropy and its modified version,for which instead online optimization schemes were not yet developed. We show,via boosting-type guarantees, that maximizing the considered objective leadsalso to the reduction of all of these entropy-based objectives. The bounds weobtain critically depend on the strong-concavity properties of theentropy-based criteria, where the mildest dependence on the number of classes(only logarithmic) corresponds to the Shannon entropy.

On the boosting ability of top-down decision tree learning algorithm for multiclass classification