Learning Markov Random Fields for Combinatorial Structures via Sampling through Lovász Local Lemma

Abstract

Learning to generate complex combinatorial structures satisfying constraints will have transformative impacts in many application domains. However, it is beyond the capabilities of existing approaches due to the highly intractable nature of the embedded probabilistic inference. Prior works spend most of the training time learning to separate valid from invalid structures but do not learn the inductive biases of valid structures. We develop NEural Lovász Sampler (Nelson), which embeds the sampler through Lovász Local Lemma (LLL) as a fully differentiable neural network layer. Our Nelson-CD embeds this sampler into the contrastive divergence learning process of Markov random fields. Nelson allows us to obtain valid samples from the current model distribution. Contrastive divergence is then applied to separate these samples from those in the training set. Nelson is implemented as a fully differentiable neural net, taking advantage of the parallelism of GPUs. Experimental results on several real-world domains reveal that Nelson learns to generate 100\% valid structures, while baselines either time out or cannot ensure validity. Nelson also outperforms other approaches in running time, log-likelihood, and MAP scores.

arXiv commentaccepted by AAAI 2023. The first two authors contribute equally

References in corpus (16)

• Auto-Encoding Variational Bayes
• Loopy Belief Propagation for Approximate Inference: An Empirical Study
• Score-Based Generative Modeling through Stochastic Differential Equations
• Generative Modeling by Estimating Gradients of the Data Distribution
• Junction Tree Variational Autoencoder for Molecular Graph Generation
• MADE: Masked Autoencoder for Distribution Estimation
• Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search
• Syntax-Directed Variational Autoencoder for Structured Data
• Survey propagation: an algorithm for satisfiability
• Learning a SAT Solver from Single-Bit Supervision
• Learning to Perform Local Rewriting for Combinatorial Optimization
• Taming the Curse of Dimensionality: Discrete Integration by Hashing and Optimization
• Learning Arithmetic Circuits
• Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs
• Augment with Care: Contrastive Learning for Combinatorial Problems
• Belief Propagation on the random $k$-SAT model