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 Dropout

Dropout has recently emerged as a powerful and simple method for training neural networks preventing co-adaptation by stochastically omitting neurons. Dropout is currently not grounded in explicit modelling assumptions which so far has precluded its adoption in Bayesian modelling. Using Bayesian entropic reasoning we show that dropout can be interpreted as optimal inference under constraints. We demonstrate this on an analytically tractable regression model providing a Bayesian interpretation of its mechanism for regularizing and preventing co-adaptation as well as its connection to other Bayesian techniques. We also discuss two general approximate techniques for applying Bayesian dropout for general models, one based on an analytical approximation and the other on stochastic variational techniques. These techniques are then applied to a Baysian logistic regression problem and are shown to improve performance as the model become more misspecified. Our framework roots dropout as a theoretically justified and practical tool for statistical modelling allowing Bayesians to tap into the benefits of dropout training.Comment: 21 pages, 3 figures. Manuscript prepared 2014 and awaiting submissio

Learning and Designing Stochastic Processes from Logical Constraints

Stochastic processes offer a flexible mathematical formalism to model and reason about systems. Most analysis tools, however, start from the premises that models are fully specified, so that any parameters controlling the system's dynamics must be known exactly. As this is seldom the case, many methods have been devised over the last decade to infer (learn) such parameters from observations of the state of the system. In this paper, we depart from this approach by assuming that our observations are {\it qualitative} properties encoded as satisfaction of linear temporal logic formulae, as opposed to quantitative observations of the state of the system. An important feature of this approach is that it unifies naturally the system identification and the system design problems, where the properties, instead of observations, represent requirements to be satisfied. We develop a principled statistical estimation procedure based on maximising the likelihood of the system's parameters, using recent ideas from statistical machine learning. We demonstrate the efficacy and broad applicability of our method on a range of simple but non-trivial examples, including rumour spreading in social networks and hybrid models of gene regulation

Raiders of the Lost Architecture: Kernels for Bayesian Optimization in Conditional Parameter Spaces

In practical Bayesian optimization, we must often search over structures with differing numbers of parameters. For instance, we may wish to search over neural network architectures with an unknown number of layers. To relate performance data gathered for different architectures, we define a new kernel for conditional parameter spaces that explicitly includes information about which parameters are relevant in a given structure. We show that this kernel improves model quality and Bayesian optimization results over several simpler baseline kernels.Comment: 6 pages, 3 figures. Appeared in the NIPS 2013 workshop on Bayesian optimizatio