Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

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

Bayesian network learning with cutting planes

The problem of learning the structure of Bayesian networks from complete discrete data with a limit on parent set size is considered. Learning is cast explicitly as an optimisation problem where the goal is to find a BN structure which maximises log marginal likelihood (BDe score). Integer programming, specifically the SCIP framework, is used to solve this optimisation problem. Acyclicity constraints are added to the integer program (IP) during solving in the form of cutting planes. Finding good cutting planes is the key to the success of the approach -the search for such cutting planes is effected using a sub-IP. Results show that this is a particularly fast method for exact BN learning

Neural Architecture Search using Deep Neural Networks and Monte Carlo Tree Search

Neural Architecture Search (NAS) has shown great success in automating the design of neural networks, but the prohibitive amount of computations behind current NAS methods requires further investigations in improving the sample efficiency and the network evaluation cost to get better results in a shorter time. In this paper, we present a novel scalable Monte Carlo Tree Search (MCTS) based NAS agent, named AlphaX, to tackle these two aspects. AlphaX improves the search efficiency by adaptively balancing the exploration and exploitation at the state level, and by a Meta-Deep Neural Network (DNN) to predict network accuracies for biasing the search toward a promising region. To amortize the network evaluation cost, AlphaX accelerates MCTS rollouts with a distributed design and reduces the number of epochs in evaluating a network by transfer learning, which is guided with the tree structure in MCTS. In 12 GPU days and 1000 samples, AlphaX found an architecture that reaches 97.84\% top-1 accuracy on CIFAR-10, and 75.5\% top-1 accuracy on ImageNet, exceeding SOTA NAS methods in both the accuracy and sampling efficiency. Particularly, we also evaluate AlphaX on NASBench-101, a large scale NAS dataset; AlphaX is 3x and 2.8x more sample efficient than Random Search and Regularized Evolution in finding the global optimum. Finally, we show the searched architecture improves a variety of vision applications from Neural Style Transfer, to Image Captioning and Object Detection.Comment: To appear in the Thirty-Fourth AAAI conference on Artificial Intelligence (AAAI-2020