Many graphical models, such as Markov Logic Networks (MLNs) with evidence, possess highly symmetric substructures but no exact symmetries. Unfortunately, there are few principled methods that exploit these symmetric substructures to perform efficient approximate inference. In this paper, we present a lifted variant of the Weighted Mini-Bucket elimination algorithm which provides a principled way to (i) exploit the highly symmetric substructure of MLN models, and (ii) incorporate high-order inference terms which are necessary for high quality approximate inference. Our method has significant control over the accuracy-time trade-off of the approximation, allowing us to generate any-time approximations. Experimental results demonstrate the utility of this class of approximations, especially in models with strong repulsive potentials.

Results are shown for two runtime limits, 2 min and 20 min. Entries are marked with '-' if the solution

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Bayesian bandit methods is necessary.

Exact inference of marginals in probabilistic graphical models (PGM) is known to be intractable, necessitating the use of approximate methods. Most of the existing variational techniques perform iterative message passing in loopy graphs which is slow to converge for many benchmarks. In this paper, we propose a new algorithm for marginal inference that is based on the incremental build-infer-approximate (IBIA) paradigm. Our algorithm converts the PGM into a sequence of linked clique tree forests (SLCTF) with bounded clique sizes, and then uses a heuristic belief update algorithm to infer the marginals. For the special case of Bayesian networks, we show that if the incremental build step in IBIA uses the topological order of variables then (a) the prior marginals are consistent in all CTFs in the SLCTF and (b) the posterior marginals are consistent once all evidence variables are added to the SLCTF. In our approach, the belief propagation step is non-iterative and the accuracy-complexity trade-off is controlled using user-defined clique size bounds. Results for several benchmark sets from recent UAI competitions show that our method gives either better or comparable accuracy than existing variational and sampling based methods, with smaller runtimes.

We study the problem of combining neural networks with symbolic reasoning. Recently introduced frameworks for Probabilistic Neurosymbolic Learning (PNL), such as DeepProbLog, perform exponential-time exact inference, limiting the scalability of PNL solutions. We introduce Approximate Neurosymbolic Inference (A-NeSI): a new framework for PNL that uses neural networks for scalable approximate inference. A-NeSI 1) performs approximate inference in polynomial time without changing the semantics of probabilistic logics; 2) is trained using data generated by the background knowledge; 3) can generate symbolic explanations of predictions; and 4) can guarantee the satisfaction of logical constraints at test time, which is vital in safety-critical applications. Our experiments show that A-NeSI is the first end-to-end method to solve three neurosymbolic tasks with exponential combinatorial scaling. Finally, our experiments show that A-NeSI achieves explainability and safety without a penalty in performance.

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational methods in approximating the Bayesian predictive distribution. For single-hidden layer ReLU BNNs, we prove a fundamental limitation in function-space of two of the most commonly used distributions defined in weight-space: mean-field Gaussian and Monte Carlo dropout. We find there are simple cases where neither method can have substantially increased uncertainty in between well-separated regions of low uncertainty. We provide strong empirical evidence that exact inference does not have this pathology, hence it is due to the approximation and not the model. In contrast, for deep networks, we prove a universality result showing that there exist approximate posteriors in the above classes which provide flexible uncertainty estimates. However, we find empirically that pathologies of a similar form as in the single-hidden layer case can persist when performing variational inference in deeper networks. Our results motivate careful consideration of the implications of approximate inference methods in BNNs.