The Theory behind Vector DB

Retrieval Augmented Generation (RAG) is a popular method for adding information to Large Language Models (LLMs), and Vector DB is the core technology used for the retrieval part of RAG. In this presentation, I will focus on the mathematical background of the approximate nearest neighbor search algorithms that are central components of Vector DB. Specifically, I will provide details on graph-based search, the most widely used family of algorithms.

The 1st Japan-Korea Workshop on Artificial Intelligence, Dec. 27, 2023 @UTokyo

Yusuke Matsui (The University of Tokyo) https://yusukematsui.me/

Transcript

1. Slides: bit.ly/41BTrrS 1 The Theory behind Vector DB Yusuke Matsui The University of Tokyo The 1st Japan-Korea Workshop on Artificial Intelligence, Dec. 27, 2023 @UTokyo
2. Slides: bit.ly/41BTrrS 2 Yusuke Matsui ✓ Computer vision ✓ Large-scale indexing ✓ Data structure + Machine learning http://yusukematsui.me Lecturer (Assistant Professor), the University of Tokyo, Japan @utokyo_bunny ARM impl. of Faiss [Matsui+, ICASSP 22] Nearest neighbor search [Ono & Matsui, ACMMM 23] @matsui528 ML-enhanced Bloom Filter [Sato & Matsui, NeurIPS 23]
3. Slides: bit.ly/41BTrrS 3 ➢ Introduction ✓ Demo: Embedding + Search + LLM ➢ The theory behind Vector DB ➢ Discussion
4. Slides: bit.ly/41BTrrS 4 ➢ Introduction ✓ Demo: Embedding + Search + LLM ➢ The theory behind Vector DB ➢ Discussion
5. Slides: bit.ly/41BTrrS 5 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “I'm sorry, but as an AI language model, I don't have information about the future events.” Ask  Want to add knowledge to LLM
6. Slides: bit.ly/41BTrrS 6 Embedding + Search + LLM（RAG; Retrieval Augmented Generation） Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021)
7. Slides: bit.ly/41BTrrS 7 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … “Chinami Yoshida¥n¥n==Personal…” Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
8. Slides: bit.ly/41BTrrS 8 𝒙1 , Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … “Chinami Yoshida¥n¥n==Personal…” Text Encoder Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
9. Slides: bit.ly/41BTrrS 9 𝒙1 , 𝒙2 , Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … “Chinami Yoshida¥n¥n==Personal…” Text Encoder Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
10. Slides: bit.ly/41BTrrS 10 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … “Chinami Yoshida¥n¥n==Personal…” Text Encoder Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
11. Slides: bit.ly/41BTrrS 11 0.23 3.15 0.65 1.43 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
12. Slides: bit.ly/41BTrrS 12 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" ChatGPT 3.5 (trained in 2021) Search “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “List of 2022 Winter Olympics medal winners…” Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
13. Slides: bit.ly/41BTrrS 13 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" Search Update “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “Who won curling gold at the 2022 Winter Olympics? Use the bellow articles: List of 2022 Winter Olympics medal winners…” “List of 2022 Winter Olympics medal winners…” ChatGPT 3.5 (trained in 2021) Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
14. Slides: bit.ly/41BTrrS 14 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" “Niklas Edin, Oskar Eriksson, …” Search Update ☺ “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “Who won curling gold at the 2022 Winter Olympics? Use the bellow articles: List of 2022 Winter Olympics medal winners…” “List of 2022 Winter Olympics medal winners…” ChatGPT 3.5 (trained in 2021) Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
15. Slides: bit.ly/41BTrrS 15 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" “Niklas Edin, Oskar Eriksson, …” Search Update ☺ “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “Who won curling gold at the 2022 Winter Olympics? Use the bellow articles: List of 2022 Winter Olympics medal winners…” “List of 2022 Winter Olympics medal winners…” ChatGPT 3.5 (trained in 2021) RAG is the simplest method to supplement knowledge to LLM Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
16. Slides: bit.ly/41BTrrS 16  DEMO1: OpenAI Playground (full managed service)
17. Slides: bit.ly/41BTrrS 17 DEMO1: OpenAI Playground (full managed service) Turn on “Retrieval” Upload Olympic information ☺ Referring uploaded information
18. Slides: bit.ly/41BTrrS 18 DEMO1: OpenAI Playground (full managed service) Turn on “Retrieval” Upload Olympic information ☺ Referring uploaded information
19. Slides: bit.ly/41BTrrS 19 DEMO2: Run on your PC (pseudo code) https://github.com/openai/openai-cookbook/blob/main/examples/Question_answering_using_embeddings.ipynb # Embed knowledge texts = [ "Chinami Yoshida¥n¥n==Persona...", "Lviv bid for the 2022 Winter...", ... ] features = encode(texts) # Embed a question question = "Who won curling gold at the 2022 Winter Olympics?" query_feature = encode(question) # Search ids = search(query_feature, features) # Add results to the question question += "Use the bellow articles" for id in ids: question += texts[id] # Ask LLM result = LLM.ask(question)
20. Slides: bit.ly/41BTrrS 20 DEMO2: Run on your PC (pseudo code) # Embed knowledge texts = [ "Chinami Yoshida¥n¥n==Persona...", "Lviv bid for the 2022 Winter...", ... ] features = encode(texts) # Embed a question question = "Who won curling gold at the 2022 Winter Olympics?" query_feature = encode(question) # Search ids = search(query_feature, features) # Add results to the question question += "Use the bellow articles" for id in ids: question += texts[id] # Ask LLM result = LLM.ask(question) ➢ エンコーダはなんでもいい ➢ 古典方式（BM25とか）もアリ ➢ モダン方式でもいい（がGPUいる） ➢ OpenAI APIでもいい（GPUいらないがお金かか る） ➢ Any encoders are fine ➢ Classic methods (e.g., BM25) is also fine ➢ Modern methods are also fine (but you need GPUs) ➢ OpenAI API is also fine (but you need to pay） How to prepare these? Search on either CPUs or GPUs （Accuracy vs Runtime vs Memory vs Money） ➢ You need GPUs to run it on your local machine. You need to prepare an executable model such as LLaMa ➢ Or, you can use OpenAI API. No need to prepare GPUs, but need to pay https://github.com/openai/openai-cookbook/blob/main/examples/Question_answering_using_embeddings.ipynb
21. Slides: bit.ly/41BTrrS 21 DEMO2: Run on your PC (pseudo code) # Embed knowledge texts = [ "Chinami Yoshida¥n¥n==Persona...", "Lviv bid for the 2022 Winter...", ... ] features = encode(texts) # Embed a question question = "Who won curling gold at the 2022 Winter Olympics?" query_feature = encode(question) # Search ids = search(query_feature, features) # Add results to the question question += "Use the bellow articles" for id in ids: question += texts[id] # Ask LLM result = LLM.ask(question) ➢ エンコーダはなんでもいい ➢ 古典方式（BM25とか）もアリ ➢ モダン方式でもいい（がGPUいる） ➢ OpenAI APIでもいい（GPUいらないがお金かか る） ➢ Any encoders are fine ➢ Classic methods (e.g., BM25) is also fine ➢ Modern methods are also fine (but you need GPUs) ➢ OpenAI API is also fine (but you need to pay） How to prepare these? Search on either CPUs or GPUs （Accuracy vs Runtime vs Memory vs Money） ➢ You need GPUs to run it on your local machine. You need to prepare an executable model such as LLaMa ➢ Or, you can use OpenAI API. No need to prepare GPUs, but need to pay ➢ “Embedding + Search” is traditional software engineering-ish ➢ API call or not ✓ Easy, but costly ✓ Reproducibility for research? https://github.com/openai/openai-cookbook/blob/main/examples/Question_answering_using_embeddings.ipynb
22. Slides: bit.ly/41BTrrS 22 ➢ Introduction ✓ Demo: Embedding + Search + LLM ➢ The theory behind Vector DB ➢ Discussion
23. Slides: bit.ly/41BTrrS 23 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" “Niklas Edin, Oskar Eriksson, …” Search Update ☺ “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “Who won curling gold at the 2022 Winter Olympics? Use the bellow articles: List of 2022 Winter Olympics medal winners…” “List of 2022 Winter Olympics medal winners…” ChatGPT 3.5 (trained in 2021) RAG is the simplest method to supplement knowledge to LLM Embedding + Search + LLM（RAG; Retrieval Augmented Generation）
24. Slides: bit.ly/41BTrrS 24 0.23 3.15 0.65 1.43 0.20 3.25 0.72 1.68 argmin 𝒒 − 𝒙𝑛 2 2 Result 𝒙1 , 𝒙2 , … , 𝒙𝑁 Texts are from: https://github.com/openai/openaicookbook/blob/main/examples/Question_answering_using_embeddings.ipynb Icon credit: https://ja.wikipedia.org/wiki/ChatGPT "Who won curling gold at the 2022 Winter Olympics?" “Niklas Edin, Oskar Eriksson, …” Search Update ☺ “Damir Sharipzyanov¥n¥n=Career…” “Lviv bid for the 2022 Winter…” … Text Encoder “Chinami Yoshida¥n¥n==Personal…” Text Encoder “Who won curling gold at the 2022 Winter Olympics? Use the bellow articles: List of 2022 Winter Olympics medal winners…” “List of 2022 Winter Olympics medal winners…” ChatGPT 3.5 (trained in 2021) RAG is the simplest method to supplement knowledge to LLM Embedding + Search + LLM（RAG; Retrieval Augmented Generation） ➢ Approx. nearest neighbor search! ➢ Accuracy vs. runtime vs. memory ➢ Vector DB??
25. Slides: bit.ly/41BTrrS 25 Three levels of technology Milvus Pinecone Qdrant ScaNN (4-bit PQ) [Guo+, ICML 2020] Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] Weaviate Vertex AI Matching Engine faiss NMSLIB hnswlib Vald ScaNN jina
26. Slides: bit.ly/41BTrrS Three levels of technology 26 Milvus Pinecone Qdrant ScaNN (4-bit PQ) [Guo+, ICML 2020] Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Weaviate Vertex AI Matching Engine NMSLIB hnswlib Vald ScaNN jina Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] faiss One library may implement multiple algorithms  “I benchmarked faiss” ☺ “I benchmarked PQ in faiss”
27. Slides: bit.ly/41BTrrS Three levels of technology 27 Milvus Pinecone Qdrant ScaNN (4-bit PQ) [Guo+, ICML 2020] Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Weaviate Vertex AI Matching Engine Vald ScaNN jina Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] faiss NMSLIB hnswlib One algorithm may be implemented in multiple libraries
28. Slides: bit.ly/41BTrrS Three levels of technology 28 Milvus Pinecone Qdrant Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] Weaviate Vertex AI Matching Engine faiss NMSLIB hnswlib Vald jina ScaNN (4-bit PQ) [Guo+, ICML 2020] ScaNN Often, one library = one algorithm
29. Slides: bit.ly/41BTrrS Three levels of technology 29 Pinecone Qdrant ScaNN (4-bit PQ) [Guo+, ICML 2020] Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Vertex AI Matching Engine NMSLIB Vald ScaNN jina Weaviate Milvus faiss hnswlib Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] One service may use some libraries … or re-implement algorithms from scratch (e.g., by Go)
30. Slides: bit.ly/41BTrrS 30 Three levels of technology Milvus Pinecone Qdrant ScaNN (4-bit PQ) [Guo+, ICML 2020] Algorithm ➢ Scientific paper ➢ Math ➢ Often, by researchers Library ➢ Implementations of algorithms ➢ Usually, a search function only ➢ By researchers, developers, etc Service (e.g., vector DB) ➢ Library + (handling metadata, serving, scaling, IO, CRUD, etc) ➢ Usually, by companies Product Quantization + Inverted Index (PQ, IVFPQ) [Jégou+, TPAMI 2011] Hierarchical Navigable Small World (HNSW) [Malkov+, TPAMI 2019] Weaviate Vertex AI Matching Engine faiss NMSLIB hnswlib Vald ScaNN jina This talk mainly focuses algorithms
31. Slides: bit.ly/41BTrrS 31 𝑁 109 106 billion-scale million-scale Locality Sensitive Hashing (LSH) Tree / Space Partitioning Graph traversal 0.34 0.22 0.68 0.71 0 1 0 0 ID: 2 ID: 123 0.34 0.22 0.68 0.71 Space partition Data compression ➢ k-means ➢ PQ/OPQ ➢ Graph traversal ➢ etc… ➢ Raw data ➢ Scalar quantization ➢ PQ/OPQ ➢ etc… Look-up-based Hamming-based Linear-scan by Asymmetric Distance … Linear-scan by Hamming distance Inverted index + data compression For raw data: Acc. ☺, Memory:  For compressed data: Acc. , Memory: ☺
32. Slides: bit.ly/41BTrrS 32 𝑁 109 106 billion-scale million-scale Locality Sensitive Hashing (LSH) Tree / Space Partitioning Graph traversal 0.34 0.22 0.68 0.71 0 1 0 0 ID: 2 ID: 123 0.34 0.22 0.68 0.71 Space partition Data compression ➢ k-means ➢ PQ/OPQ ➢ Graph traversal ➢ etc… ➢ Raw data ➢ Scalar quantization ➢ PQ/OPQ ➢ etc… Look-up-based Hamming-based Linear-scan by Asymmetric Distance … Linear-scan by Hamming distance Inverted index + data compression For raw data: Acc. ☺, Memory:  For compressed data: Acc. , Memory: ☺ De facto standard! Will explain
33. Slides: bit.ly/41BTrrS 33 Graph search ➢ De facto standard if all data can be loaded on memory ➢ Fast and accurate for real-world data Images are from [Malkov+, Information Systems, 2013] ➢ Traverse graph towards the query ➢ Seems intuitive, but not so much easy to understand ➢ Review the algorithm carefully
34. Slides: bit.ly/41BTrrS 34 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M ➢ Given a query vector Candidates (size = 3) Close to the query Name each node for explanation
35. Slides: bit.ly/41BTrrS 35 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M ➢ Given a query vector ➢ Start from an entry point (e.g., ) Candidates (size = 3) Close to the query M
36. Slides: bit.ly/41BTrrS 36 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M ➢ Given a query vector ➢ Start from an entry point (e.g., ). Record the distance to q. Candidates (size = 3) Close to the query M M 23.1
37. Slides: bit.ly/41BTrrS 37 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query 23.1 M
38. Slides: bit.ly/41BTrrS 38 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query M 23.1 1st iteration
39. Slides: bit.ly/41BTrrS 39 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M ➢ Pick up the unchecked best candidate ( ) Candidates (size = 3) Close to the query M M 23.1 Best Best
40. Slides: bit.ly/41BTrrS 40 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M ➢ Pick up the unchecked best candidate ( ). Check it. Candidates (size = 3) Close to the query M M 23.1 Best Best check!
41. Slides: bit.ly/41BTrrS 41 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query 23.1 Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. Best M M check!
42. Slides: bit.ly/41BTrrS 42 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query 23.1 Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. N M M check!
43. Slides: bit.ly/41BTrrS 43 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. N M J 11.1 N 15.3 K 19.4 M 23.1 check!
44. Slides: bit.ly/41BTrrS 44 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. N M J 11.1 N 15.3 K 19.4 M 23.1
45. Slides: bit.ly/41BTrrS 45 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) N M J 11.1 N 15.3 K 19.4 M 23.1
46. Slides: bit.ly/41BTrrS 46 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) N M J 11.1 N 15.3 K 19.4
47. Slides: bit.ly/41BTrrS 47 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4
48. Slides: bit.ly/41BTrrS 48 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4 2nd iteration
49. Slides: bit.ly/41BTrrS 49 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4 ➢ Pick up the unchecked best candidate ( ) J Best Best
50. Slides: bit.ly/41BTrrS 50 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4 ➢ Pick up the unchecked best candidate ( ). Check it. J Best Best check!
51. Slides: bit.ly/41BTrrS 51 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. J Best Best check!
52. Slides: bit.ly/41BTrrS 52 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N J 11.1 N 15.3 K 19.4 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. J Best 13.2 9.7 check! Already visited
53. Slides: bit.ly/41BTrrS 53 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. J Best 13.2 9.7 J 11.1 N 15.3 B 2.3 G 3.5 I 9.7 F 10.2 L 13.2 check! Already visited
54. Slides: bit.ly/41BTrrS 54 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. J Best J 11.1 N 15.3 B 2.3 G 3.5 I 9.7 F 10.2 L 13.2
55. Slides: bit.ly/41BTrrS 55 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) J Best J 11.1 N 15.3 B 2.3 G 3.5 I 9.7 F 10.2 L 13.2
56. Slides: bit.ly/41BTrrS 56 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) J Best B 2.3 G 3.5 I 9.7
57. Slides: bit.ly/41BTrrS 57 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7
58. Slides: bit.ly/41BTrrS 58 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7 3rd iteration
59. Slides: bit.ly/41BTrrS 59 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7 Best Best ➢ Pick up the unchecked best candidate ( ) B
60. Slides: bit.ly/41BTrrS 60 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7 Best Best ➢ Pick up the unchecked best candidate ( ). Check it. B check!
61. Slides: bit.ly/41BTrrS 61 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7 Best Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. B check!
62. Slides: bit.ly/41BTrrS 62 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N B 2.3 G 3.5 I 9.7 Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. B 0.5 2.1 check!
63. Slides: bit.ly/41BTrrS 63 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N Best ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. B 0.5 2.1 C 0.5 D 2.1 A 3.6 B 2.3 G 3.5 I 9.7 check!
64. Slides: bit.ly/41BTrrS 64 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. B C 0.5 D 2.1 A 3.6 B 2.3 G 3.5 I 9.7 Best
65. Slides: bit.ly/41BTrrS 65 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) B C 0.5 D 2.1 A 3.6 B 2.3 G 3.5 I 9.7 Best
66. Slides: bit.ly/41BTrrS 66 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) B C 0.5 D 2.1 B 2.3 Best
67. Slides: bit.ly/41BTrrS 67 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3
68. Slides: bit.ly/41BTrrS 68 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 4th iteration
69. Slides: bit.ly/41BTrrS 69 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). C Best Best
70. Slides: bit.ly/41BTrrS 70 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. C Best Best check!
71. Slides: bit.ly/41BTrrS 71 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. C Best Best check!
72. Slides: bit.ly/41BTrrS 72 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. C Best check! Already visited Already visited Already visited Already visited
73. Slides: bit.ly/41BTrrS 73 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) C Best
74. Slides: bit.ly/41BTrrS 74 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3
75. Slides: bit.ly/41BTrrS 75 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 5th iteration
76. Slides: bit.ly/41BTrrS 76 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). D Best Best
77. Slides: bit.ly/41BTrrS 77 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. D Best Best check!
78. Slides: bit.ly/41BTrrS 78 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. D Best Best check!
79. Slides: bit.ly/41BTrrS 79 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. D Best check! Already visited Already visited
80. Slides: bit.ly/41BTrrS 80 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. D Best check! Already visited Already visited H 3.9
81. Slides: bit.ly/41BTrrS 81 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. D Best H 3.9
82. Slides: bit.ly/41BTrrS 82 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) D Best H 3.9
83. Slides: bit.ly/41BTrrS 83 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ Pick up the unchecked best candidate ( ). Check it. ➢ Find the connected points. ➢ Record the distances to q. ➢ Maintain the candidates (size=3) D Best
84. Slides: bit.ly/41BTrrS 84 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ All candidates are checked. Finish. ➢ Here, is the closet to the query ( ) C
85. Slides: bit.ly/41BTrrS 85 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 C Final output 1: Candidates ➢ You can pick up topk results ➢ All candidates are checked. Finish. ➢ Here, is the closet to the query ( )
86. Slides: bit.ly/41BTrrS 86 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 C Final output 1: Candidates ➢ You can pick up topk results ➢ All candidates are checked. Finish. ➢ Here, is the closet to the query ( ) Final output 2: Checked items ➢ i.e., search path
87. Slides: bit.ly/41BTrrS 87 Search Images are from [Malkov+, Information Systems, 2013] A B C D E F G H I J K L N M Candidates (size = 3) Close to the query N C 0.5 D 2.1 B 2.3 ➢ All candidates are checked. Finish. ➢ Here, is the closet to the query ( ) C Final output 1: Candidates ➢ You can pick up topk results Final output 2: Checked items ➢ i.e., search path Final output 3: Visit flag ➢ For each item, visited or not
88. Slides: bit.ly/41BTrrS 88 Observation: runtime ➢ Item comparison takes time; 𝑂 𝐷 ➢ The overall runtime ~ #item_comparison ∼ length_of_search_path * average_outdegree 𝒒 ∈ ℝ𝐷 𝒙13 ∈ ℝ𝐷 start query start query start query 1st path 2nd path 3rd path 2.1 1.9 outdegree = 1 outdegree = 2 outdegree = 2 #item_comparison = 3 * (1 + 2 + 2)/3 = 5 2.4
89. Slides: bit.ly/41BTrrS 89 A D C B query Observation: candidate size E start Candidates (size = 1) C A D C B query E start Candidates (size = 3) C D E size = 1: Greedy search size > 1: Beam search ➢ Larger candidate size, better but slower results ➢ Online parameter to control the trade-off ➢ Called “ef” in HNSW Fast. But stuck in a local minimum Slow. But find a better solution
90. Slides: bit.ly/41BTrrS 90 Graph search algorithms Images are from an excellent survey paper [Wang+, VLDB 2021] ➢ Lots of algorithms ➢ Ours also! Ono & Matsui, “Relative NN-Descent”, ACMMM 23 ➢ The basic structure is same: (1) designing a good graph + (2) beam search
91. Slides: bit.ly/41BTrrS 91 Just NN? Vector DB? ➢ Vector DB companies say “Vector DB is cool” ➢ My own idea: ➢ Which vector DB? ➡ No conclusions! ✓ https://weaviate.io/blog/vector-library-vs-vector-database ✓ https://codelabs.milvus.io/vector-database-101-what-is-a-vector-database/index#2 ✓ https://zilliz.com/learn/what-is-vector-database Try the simplest numpy–only search Slow? Try fast algorithm such as HNSW in faiss Try Vector DB If speed is the only concern, just use libraries
92. Slides: bit.ly/41BTrrS 92 ➢ Introduction ✓ Demo: Embedding + Search + LLM ➢ The theory behind Vector DB ➢ Discussion
93. Slides: bit.ly/41BTrrS 93 Recent trends ➢ “Embedding + Search” is too naïve? ✓ “A similar sentence to the question” might not be useful at all. ➢ Smarter search? ✓ Classic technologies have been revisited! ✓ E.g., Query expansion  Recall [Chum+, ICCV 2007]. They applied query expansion, which was a classic IR technique, to image retrieval.  Revisit repeats! ➢ Directions ✓ Solve in a machine-learning way? E.g., training a special embedding space. ✓ Solve by data-structures? E.g., data structure with a special operations.
94. Slides: bit.ly/41BTrrS 94 Recent trends ➢ “Embedding + Search” is too naïve? ✓ “A similar sentence to the question” might not be useful at all. ➢ Smarter search? ✓ Classic technologies have been revisited! ✓ E.g., Query expansion  Recall [Chum+, ICCV 2007]. They applied query expansion, which was a classic technique, to image retrieval.  Revisit repeats! ➢ Directions ✓ Solve in a machine-learning way? E.g., training a special embedding space. ✓ Solve by data-structures? E.g., data structure with a special operations.
95. Slides: bit.ly/41BTrrS 95 Recent trends ➢ “Embedding + Search” is too naïve? ✓ “A similar sentence to the question” might not be useful at all. ➢ Smarter search? ✓ Classic technologies have been revisited! ✓ E.g., Query expansion  Recall [Chum+, ICCV 2007]. They applied query expansion, which was a classic IR technique, to image retrieval.  Revisit repeats! ➢ Directions ✓ Solve in a machine-learning way? E.g., training a special embedding space. ✓ Solve by data-structures? E.g., data structure with a special operations.
96. Slides: bit.ly/41BTrrS 96 From the perspective of Nearest Neighbor Search ➢ CPU? GPU? Managed services? ✓ Trade-off! ✓ NVIDIA pushing GPU-search (RAFT&CAGRA [Ootomo+, arXiv 23]) ✓ Can we spend money to search? (Not training a model) ➢ Next paradigm ✓ Graph-based search is mature. ✓ We need breakthrough idea? ➢ Not trivial：Dimensionality (𝐷) is now 1000 (not 100) ✓ If 𝐷 > 100, it’s often said that we must PCA it to 𝐷 = 100. ✓ Because of LLM API, we need to handle 𝐷 = 1000 anyway.
97. Slides: bit.ly/41BTrrS 97 Use LLM as a building block ➢ The era of using LLMs as functions in pseudo code has already arrived. ➢ Should we run something locally or API call? A choice that has never been made before. ➢ How to think about "Reproducibility" in the LLM and API Era ➢ Need to be able to roughly estimate what is possible solutions to a problem, and at what cost, and what will be obtained. On-premises own server Cloud (e.g., AWS EC2) API Call ☺Free to use Initial cost maintenance ☺Good balance Depends on cloud ☺Super cheap Not good at large-processing Communication cost? New paradigm
98. Slides: bit.ly/41BTrrS 98 Reference ➢ Survey for NN search: CVPR 20 Tutorial ➢ SOTA graph-based search: CVPR 23 Tutorial ➢ Good tutorial for RAG [Asai+, ACL 2023 Tutorial]
99. Slides: bit.ly/41BTrrS 99 ◼ [Jégou+, TPAMI 2011] H. Jégou+, “Product Quantization for Nearest Neighbor Search”, IEEE TPAMI 2011 ◼ [Guo+, ICML 2020] R. Guo+, “Accelerating Large-Scale Inference with Anisotropic Vector Quantization”, ICML 2020 ◼ [Malkov+, TPAMI 2019] Y. Malkov+, “Efficient and Robust Approximate Nearest Neighbor search using Hierarchical Navigable Small World Graphs,” IEEE TPAMI 2019 ◼ [Malkov+, IS 13] Y, Malkov+, “Approximate Nearest Neighbor Algorithm based on Navigable Small World Graphs”, Information Systems 2013 ◼ [Fu+, VLDB 19] C. Fu+, “Fast Approximate Nearest Neighbor Search With The Navigating Spreading-out Graphs”, 2019 ◼ [Wang+, VLDB 21] M. Wang+, “A Comprehensive Survey and Experimental Comparison of Graph-Based Approximate Nearest Neighbor Search”, VLDB 2021 ◼ [Iwasaki+, arXiv 18] M. Iwasaki and D. Miyazaki, “Optimization if Indexing Based on k-Nearest Neighbor Graph for Proximity Search in High-dimensional Data”, arXiv 2018 ◼ [Ootomo+, arXiv 23] H. Ootomo+, “CAGRA: Highly Parallel Graph Construction and Approximate Nearest Neighbor Search for GPUs”, arXiv 2023 ◼ [Chum+, ICCV 07] O. Chum+, “Total Recall: Automatic Query Expansion with a Generative Feature Model for Object Retrieval”, ICCV 2007 ◼ [Pinecone] https://www.pinecone.io/ ◼ [Milvus] https://milvus.io/ ◼ [Qdrant] https://qdrant.tech/ ◼ [Weaviate] https://weaviate.io/ ◼ [Vertex AI Matching Engine] https://cloud.google.com/vertex-ai/docs/matching-engine ◼ [Vald] https://vald.vdaas.org/ ◼ [Modal] https://modal.com/ Reference