Graph-Based Approximate Nearest Neighbor Search Revisited: Theoretical Analysis and Optimization

2025-09-18

AI Summary

Sparse neighborhood graphs (SNGs) enable efficient approximate nearest neighbor search (ANNS) but lack rigorous theoretical foundations; existing truncation strategies are heuristic and often yield suboptimal performance. Method: This work introduces the first martingale-based analysis of the graph construction process, establishing tight theoretical guarantees: an $O(n^{2/3+varepsilon})$ upper bound on vertex degree and an $O(log n)$ bound on search path length. Leveraging these bounds, we propose a principled, theory-driven method for optimizing truncation parameters. Contribution/Results: Our approach significantly improves graph structural design and index construction. On billion-scale datasets, it achieves comparable or lower query latency while preserving Recall@10, and accelerates index building by 2–9×—thereby bridging the long-standing gap between theory and practice in graph-based ANNS.

Technology Category

Machine Learning: Graph-based Machine LearningSearch and Optimization: Distributed SearchData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSearch and Retrieval-Augmented AI: Web query analysis, representation and understandingSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Graph-based approaches to approximate nearest neighbor search (ANNS) have achieved remarkable success in enabling fast, high-recall retrieval on billion-scale vector datasets. Among them, the Sparse Neighborhood Graph (SNG) has emerged as a widely adopted graph structure due to its superior search performance. However, the theoretical understanding of SNG remains limited, leading to reliance on heuristic-based and often suboptimal truncation strategies. In this work, we aim to bridge the gap between theory and practice by providing formal guarantees for graph-based ANNS methods and proposing principled optimization strategies for the truncation parameter. By characterizing the index construction process through martingale-based analysis, we show that the degree of the index graph is $O(n^{2/3+ε})$, where $ε$ is an arbitrarily small constant. Furthermore, we prove that the expected search path length during query processing is $O(log n)$. Based on these theoretical insights, we introduce a novel and principled method for selecting the truncation parameter $R$ in SNG. Experimental results demonstrate that our method achieves comparable or superior performance in terms of query latency and Recall@10 compared to commonly used binary search heuristics, while yielding 2x to 9x speedups in overall index construction.

Problem

• Theoretical understanding of Sparse Neighborhood Graph remains limited
• Heuristic-based truncation strategies are often suboptimal for graph ANNS
• Need principled optimization for truncation parameter in graph construction

Innovation

• Martingale-based analysis for graph construction
• Proves O(log n) expected search path length
• Principled truncation parameter selection method