Convex Structure Learning for Bayesian Networks: Polynomial Feature Selection and Approximate Ordering
This work addresses the computationally hard problem of optimizing variable orders in Bayesian networks for researchers in machine learning and statistics, though it appears incremental as it builds on existing convex relaxation techniques.
The authors tackled the problem of learning Bayesian network structures and parameters efficiently without restricting parent set sizes, by introducing a convex relaxation that yields an optimal 'soft' ordering in polynomial time, and experimentally compared it against standard methods to reduce local minima effects.
We present a new approach to learning the structure and parameters of a Bayesian network based on regularized estimation in an exponential family representation. Here we show that, given a fixed variable order, the optimal structure and parameters can be learned efficiently, even without restricting the size of the parent sets. We then consider the problem of optimizing the variable order for a given set of features. This is still a computationally hard problem, but we present a convex relaxation that yields an optimal 'soft' ordering in polynomial time. One novel aspect of the approach is that we do not perform a discrete search over DAG structures, nor over variable orders, but instead solve a continuous relaxation that can then be rounded to obtain a valid network structure. We conduct an experimental comparison against standard structure search procedures over standard objectives, which cope with local minima, and evaluate the advantages of using convex relaxations that reduce the effects of local minima.