Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

This is a talk given by Simge Kucukyavuz as part of the Online Seminar Series Machine Learning NeEDS Mathematical Optimization

Abstract

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions.  However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear big-M constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints.  Our numerical results demonstrate the effectiveness of the proposed approaches.

More information

Link to the talk: https://eu.bbcollab.com/collab/ui/session/guest/cff4e22f2bc746fa9e7aa9eba4cb7e4c

Organisers

The organizers of the Online Seminar Series Machine Learning NeEDS Mathematical Optimization
Emilio Carrizosa, IMUS-Instituto de Matemáticas de la Universidad de Sevilla
Thomas Halskov, Copenhagen Business School
Kseniia Kurishchenko, Copenhagen Business School
Cristina Molero-Río, IMUS-Instituto de Matemáticas de la Universidad de Sevilla
Jasone Ramírez-Ayerbe, IMUS-Instituto de Matemáticas de la Universidad de Sevilla
Dolores Romero Morales, Copenhagen Business School