Automorphism Groups of Graphical Models and Lifted Variational Inference
This work addresses the problem of computational efficiency in probabilistic inference for machine learning, offering a novel mathematical framework that is incremental but improves upon existing variational methods.
The authors formalized the concept of symmetry in probabilistic models using automorphism groups, enabling lifted inference by grouping variables and features into orbits to compute marginals efficiently. They applied this framework to variational approximations for MAP inference, achieving state-of-the-art performance with better objective values than local approximations while maintaining efficiency.
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference in the general exponential family. Its group action partitions the set of random variables and feature functions into equivalent classes (called orbits) having identical marginals and expectations. Then the inference problem is effectively reduced to that of computing marginals or expectations for each class, thus avoiding the need to deal with each individual variable or feature. We demonstrate the usefulness of this general framework in lifting two classes of variational approximation for MAP inference: local LP relaxation and local LP relaxation with cycle constraints; the latter yields the first lifted inference that operate on a bound tighter than local constraints. Initial experimental results demonstrate that lifted MAP inference with cycle constraints achieved the state of the art performance, obtaining much better objective function values than local approximation while remaining relatively efficient.