Title: Efficient Training and Inference of Hypergraph Reasoning Networks. Abstract: We study the problem of hypergraph reasoning in large domains, e.g., predicting the relationship between several entities based on the input facts. We observe that in logical reasoning, logical rules (e.g., my parent's parent is my grandparent) usually apply locally (e.g., only three people are involved in a grandparent rule), and sparsely (e.g., the grandparent relationship is sparse across all pairs of people in the world). Inspired by these observations, we propose Sparse and Local Neural Logic Machines (SpaLoc), a structured neural network for hypergraph reasoning. To leverage the sparsity in hypergraph neural networks, SpaLoc represents the grounding of relationships such as parent and grandparent as sparse tensors and uses neural networks and finite-domain quantification operations to infer new facts based on the input. We further introduce a sparsification loss to regularize the number of hyperedges in intermediate layers of a SpaLoc model. To enable training on large-scale graphs such as real-world knowledge graphs, SpaLoc makes training and inference-time sub-sampling of the input graphs. To remedy the information loss in sampled sub-graphs, we propose a novel sampling and label calibration paradigm based on an information-theoretic measure information sufficiency. Our SpaLoc shows superior accuracy and efficiency on synthetic datasets compared with prior art and achieves state-of-the-art performance on several real-world knowledge graph reasoning benchmarks.