Online Forest Density Estimation

Frederic M Koriche

Abstract: Online density estimation is the problem of predicting a sequence of outcomes, revealed one at a time, almost as well as the best expert chosen from a reference class of probabilistic models. The performance of each expert is measured with the log-likelihood loss. The class of experts examined in this paper is the family of discrete, acyclic graphical models, also known as Markov forests. By coupling Bayesian mixtures with symmetric Dirichlet priors for parameter learning, and a variant of “Follow the Perturbed Leader” strategy for structure learning, we derive an online forest density estimation algorithm that achieves a regret of $\tilde O( \sqrt T)$, with a per-round time complexity that is quasi-quadratic in the input dimension. Using simple and flexible update rules, this algorithm can be easily adapted to predict with Markov trees or mixtures of Markov forests. Empirical results indicate that our online algorithm is a practical alternative to the state-of-the-art batch algorithms for learning tree-structured graphical models.