We consider the problem of estimating the conditional probability of a label in time O(logn), where n is the number of possible labels. We analyze a natural reduction of this problem to a set of binary regression problems organized in a tree structure, proving a regret bound that scales with the depth of the tree. Motivated by this analysis, we propose the first online algorithm which provably constructs a logarithmic depth tree on the set of labels to solve this problem. We test the algorithm empirically, showing that it works succesfully on a dataset with roughly 106 labels.
@article{arxiv.0903.4217,
title = {Conditional Probability Tree Estimation Analysis and Algorithms},
author = {Alina Beygelzimer and John Langford and Yuri Lifshits and Gregory Sorkin and Alex Strehl},
journal= {arXiv preprint arXiv:0903.4217},
year = {2009}
}