Modelling problems containing a mixture of Boolean and numerical variables is a long-standing interest of Artificial Intelligence. However, performing inference and learning in hybrid domains is a particularly daunting task. The ability to model this kind of domains is crucial in "learning to design" tasks, that is, learning applications where the goal is to learn from examples how to perform automatic de novo design of novel objects. In this paper we present Structured Learning Modulo Theories, a max-margin approach for learning in hybrid domains based on Satisfiability Modulo Theories, which allows to combine Boolean reasoning and optimization over continuous linear arithmetical constraints. The main idea is to leverage a state-of-the-art generalized Satisfiability Modulo Theory solver for implementing the inference and separation oracles of Structured Output SVMs. We validate our method on artificial and real world scenarios.
Relational Hybrid Models @cite_23 (RHM) extend Relational Continuous Models to represent combinations of discrete and continuous distributions. The authors present a family of lifted variational algorithms for performing efficient inference, showing substantial improvements over their ground counterparts. As for most hybrid SRL approaches which will be discussed further on, the authors focus on efficiently computing probabilities rather than efficiently finding optimal configurations. Exact inference, hard constraints and theories like algebra over integers, which are naturally handled by our LMT framework, are all out of the scope of these approaches. Nonetheless, lifted inference is a powerful strategy to scale up inference and equipping OMT and SMT tools with lifting capabilities is a promising direction for future improvements.
@cite_23
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Hybrid continuous-discrete models naturally represent many real-world applications in robotics, finance, and environmental engineering. Inference with large-scale models is challenging because relational structures deteriorate rapidly during inference with observations. The main contribution of this paper is an efficient relational variational inference algorithm that factors largescale probability models into simpler variational models, composed of mixtures of iid (Bernoulli) random variables. The algorithm takes probability relational models of largescale hybrid systems and converts them to a close-to-optimal variational models. Then, it efficiently calculates marginal probabilities on the variational models by using a latent (or lifted) variable elimination or a lifted stochastic sampling. This inference is unique because it maintains the relational structure upon individual observations and during inference steps.