Introduction to HNSW: Hierarchical Navigable Small World

Introduction

AI innovation is going on at breakneck pace. One of many frontiers of this innovation is the vector serps. What are these serps, you ask? In easy phrases, they assist in coaching massive language fashions (LLMs) by going by massive datasets and choosing out what’s related. Now, there are lots of alternative ways by which this indexing is completed in vector databases. Amongst them, Hierarchical Navigable Small World (HNSW) stands out for being performant and scalable. All the key vector shops present HNSW as an indexing technique. It’s quick, environment friendly, strong, and dependable. So, on this article, we are going to break down the inside workings of HNSW and study what makes it so quick.

Studying Targets

• Perceive what embeddings and vector databases are.
• Get conversant in the alternative ways of indexing in vector databases.
• Study what HNSW is and the way it works.
• Perceive HNSWlib, a header-only HNSW implementation.

This text was revealed as part of the Information Science Blogathon.

What are Embeddings?

Embeddings are vector representations of information (textual content, picture) in a high-dimensional vector house.

Two semantically associated information might be in proximity within the vector house, whereas dissimilar information might be far-off. In different phrases, the phrase embeddings for Messi and soccer might be shut collectively within the embedding house, whereas the phrase embeddings for soccer and Joe Biden might be far aside within the embedding house.

The size of vectors can vary from just a few hundred to hundreds or extra. This makes it troublesome to retailer, question, and search. However each Retrieval Augmented Era (RAG) based mostly utility requires high-speed search and querying of information embeddings. That is the place Vector Databases come into the image.

What are Vector Databases?

Simply as conventional databases goal to retailer structured and unstructured information, vector databases retailer, search, and question high-dimensional vector embeddings. They supply user-friendly interfaces to work together with embeddings and their related information. Vector databases will not be basically totally different from conventional databases. Vector databases use conventional databases to retailer serialized embeddings. For instance, Chroma makes use of SQLite as in-memory storage, and Pgvector makes use of the Postgres database to retailer embeddings and their related metadata. The factor that separates a standard database from a vector database is the underlying indexing algorithm.

Indexing in Vector Databases

Indexing refers back to the strategy of organizing high-dimensional vectors in a approach that gives environment friendly querying of nearest-neighbor vectors.

That is essentially the most essential a part of constructing any vector database. These indexes allow quick and environment friendly querying of high-dimensional embeddings. There are a number of indexing strategies to create vector indices, reminiscent of:

• Linear search Algorithm (Flat Indexing): It is a linear search algorithm, which suggests it should evaluate the question vector with each different vector saved within the database. That is the best technique on the market and works nicely with small datasets.
• Cluster-based algorithm (IVF): Inverted File is a cluster-based indexing method. It makes use of k-means clustering to cluster all of the vectors. When a question vector is supplied, it calculates the space between the question vector and the centroids of every cluster. And begins looking for the closest neighbors within the cluster with the centroid closest to the question vector. This considerably reduces question time.
• Quantization (Scalar and Product Quantization): The quantization method includes decreasing the reminiscence footprint of huge embeddings by decreasing their precision.
• Graph-based (HNSW): The commonest indexing technique. It makes use of hierarchical graph structure to index vectors. And that is what we’re going to discover.

Understanding HNSW

Massive language fashions (LLMs) have gotten more and more well-liked, and plenty of organizations wish to implement them of their product stacks. Nevertheless, there’s a problem to doing this: LLMs have a restricted context window. A context window is the variety of tokens an AI mannequin can ingest. For instance, the GPT 3.5 turbo has a context size of 4096. That is the place vector search databases shine. As an alternative of throwing the whole ebook into the context of LLM, we discover essentially the most related elements and feed them to the LLM to get exact outcomes.

Now, amongst all of the alternative ways of indexing in vector databases mentioned above, HNSW is essentially the most strong and scalable. This makes it essentially the most broadly used indexing technique as nicely. HNSW is fashioned by combining two algorithms: skip listing and navigable small world. To know HNSW, we have to know these algorithms. So, let’s dive in.

Skip Record

Because the identify suggests, the Skip listing relies on the linked listing information construction, or we will say it’s an extension of the linked listing information construction. It was invented by David Pugh in 1990 as a sooner different to linked lists.

Why will we even want a skip listing? A linked listing has an O(n) search time complexity. This might not be supreme for real-world use instances the place pace of execution is paramount. So, because of this we might have a extra environment friendly linked-list algorithm.

The skip lists have an anticipated time complexity of O(log n). It performs significantly better at random entry than linked lists. Because it has a layered construction with a number of nodes at every layer, the worst-case house complexity is O(n log n), the place n is the variety of nodes on the bottom-most stage.

How Does Skip Record Work?

A Skip listing maintains a layered linked structure the place the highest layer has the longest hyperlinks between parts. This reduces exponentially as we transfer down the layers.

Within the above image, the bottom-most layer is a whole linked listing. As we transfer up, the variety of nodes reduces by half in every layer. The construction is named skip lists, as the upper layers allow you to skip nodes whereas traversing.

Think about the next instance.

When looking for okay:

• if okay = goal ingredient
• if okay >= transfer proper
• if okay < transfer down

We begin from the highest left nook and transfer proper till we discover okay or a quantity lower than okay. We descend to the layer under and proceed the method till we attain okay. The search complexity is O(log n) as we skip half the objects at every stage.

Whereas random entry is quicker, insertion and deletion are slower as they add extra overhead for updating and deleting on a number of layers.

Whereas insertion, we begin from the underside listing and add the node on the applicable place. As skip lists preserve a hierarchical construction, we have to decide if the node seems at the next stage. The method is random, like a coin toss. The chance of a node showing in its quick higher layer is 0.5. In a really perfect skip listing, the variety of nodes on layer-1 might be ~n/2, and in layer-2 ~n/4, the place n is the variety of nodes on the bottom-most layer or the whole linked listing.

Think about the next instance.

We discover the perfect place for insertion and insert the node on the backside stage. We then resolve if the node seems on the higher stage based mostly on a random binary final result (heads or tail). In an ideal skip listing, we get a balanced distribution of nodes in every stage.

Deletion occurs equally. Discover the goal quantity and delete the node. If the ingredient is there on the next layer, delete it and replace the linked listing.

Navigable Small World (NSW)

Navigable Small World is a graph-based algorithm for locating approximate nearest neighbors. The info factors in a graph are known as nodes. Every node is linked to a set set of connections which might be shut to one another.

It is a grasping algorithm. It begins at a pre-defined level within the graph and selects nodes nearer to the goal node. The gap between the nodes might be measured by Euclidean or Cosine similarity. This course of is repeated till it reaches the closest neighbors of the goal node.

The Navigable Small World algorithm could be very environment friendly and simply deployable. It really works nicely for datasets starting from just a few hundred to hundreds. After that, it tends to carry out worse. It suffers from pre-mature termination when it cannot discover a higher neighbor than the present one, because it makes use of solely native data to seek out the closest neighbor.

Throughout insertion, we traverse the graph to seek out the closest neighbors and join them to the node x.

As in vector databases, we have to cope with a number of thousands and thousands of embedding information. Therefore, we’d like a greater algorithm that scales nicely and provides higher searchability. Whereas NSW performs nicely sufficient for small datasets, we’d like a fair higher algorithm to deal with 100s of thousands and thousands or billions of information factors. That is the place HNSW comes into the image.

Hierarchical Navigable Small World (HNSW)

HNSW extends NSW by incorporating the hierarchical structure of Skip Lists. This eliminated the scalability bottleneck of NSW. Like Skip Lists, HNSW creates the multi-layer construction of NSWs as an alternative of linked lists. Like Skip Lists, the topmost layer may have fewer information factors with the longest connections. The variety of parts will increase as we transfer down the hierarchy. On the bottom-most stage, now we have all the info factors. Like skip lists, the chance of the existence of a component exponentially decreases as we transfer up within the hierarchy.

However how will we search in HNSW?

Looking out in HNSW

Now recall each the skip listing and NSW. Within the skip listing, we begin on the uppermost layer, and in NSW, we begin at a pre-defined level. In HNSW, we begin from a pre-defined level on the uppermost layer of the hierarchy and greedily traverse the graph to seek out the closest ingredient to the goal information level in that layer. As soon as we attain the closest node, we descend to the layer under and repeat the method till “Okay” nearest neighbors of the goal node are discovered. See under image

Insertion and Deletion in HNSW

Insertion in HNSW follows the identical precept as that of Skip lists. We traverse the layers, discovering the closest neighbors of the ingredient. Then, we transfer down and repeat the identical course of till we discover all the closest neighbors on the underside layer.

The subsequent process is to find out the bi-directional hyperlinks to the inserted ingredient. It’s often decided by a predefined parameter m. We join the m variety of nearest neighbors to the inserted node. This is without doubt one of the methods to find out connections to an inserted node. Different heuristics can be used. For example, as an alternative of solely connecting to the closest neighbor nodes of the identical area, we additionally join the inserted node to the closest node of the closest area to type a better-connected graph.

As with the skip lists, the chance of a node showing within the larger layers is set randomly. The perform for it’s ground(-ln(rand(0, 1))), the place rand(0, 1) is a random quantity sampled from a uniform distribution between (0, 1].

Deletion follows an identical method. We begin with the highest layer and delete the goal because it seems until the underside layer.

Complexities in HNSW

The time complexities of looking, insertion, and deleting in HNSW rely upon a number of components, together with the peak of the structure, the variety of neighboring nodes per node, and the space metric. However on common, looking, insertion, and deletion have O(log n) time complexity. The development of the HNSW might be costly. We have to insert n variety of nodes with O(log n) complexity. So, the general time complexity of graph development might be O(n log n).

Vector databases are constructed to deal with tons of of thousands and thousands of embeddings. Indexing such an quantity of information requires a extremely environment friendly, strong, and scalable algorithm. HNSW ticks all of the bins.

Cons of HNSW

Though the looking, insertion, and deletion are sooner in HNSW, there are just a few trade-offs it’s essential know earlier than going with HNSW. Right here are some things to remember earlier than implementing HNSW.

• Greater Reminiscence footprint: HNSW maintains a hierarchical construction of embeddings, which considerably will increase reminiscence consumption in comparison with algorithms like NSW. This may be problematic for resource-constraint gadgets.
• Parameter Tuning: HNSW has totally different tunable parameters. Cautious parameter tuning is required to enhance efficiency.
• Problem: Implementing HNSW from scratch can get difficult. Most vector databases use trusted pre-built options reminiscent of FAISS or HNSWlib.

HNSWlib is a header-only C++ implementation of the HNSW algorithm with Python bindings. It was written by the writer of the HNSW paper, Yury Malkov. It is a bare-bone implementation of the algorithm.

So, let’s get into it.

You possibly can set up HNSWlib with any Python package deal supervisor.

pip set up hnswlib

Declare and Initialize HNSW index.

import hnswlib
import numpy as np
import pickle

dim = 16
num_elements = 100

hnsw_index = hnswlib.Index(house="l2", dim = dim) #declare Index
hnsw_index.init_index(max_elements = num_elements, ef_construction = 200, M = 16)

• The house parameter is the space metric that might be used to calculate the space between nodes. Python implementation helps squared L2, Cosine, and dot-product.
• The dim parameter is the dimension of embedding vectors
• The init_index technique initializes the index.
• ef_construction defines the development time/accuracy trade-off.
• M is the variety of bi-directional hyperlinks created throughout the insertion of a node.

Now that we created the indexes let’s add just a few vectors.

data1 = np.float32(np.random.random((num_elements, dim)))
ids1 = np.arange(num_elements)
data2 = np.float32(np.random.random((100, dim)))
ids2 = np.arange(100)
data3 = np.float32(np.random.random((100, dim)))
ids3 = np.arange(100)

hnsw_index.add_items(data1, ids1)
hnsw_index.add_items(data2, ids2)

hnsw_index.set_ef(50) #set question time pace/accuracy trade-off
hnsw_index.set_num_threads(4) #units variety of threads throughout batch development

Now, let’s see how we will question okay’s approximate nearest neighbor.

labels, distances = p.knn_query(information, okay = 1)

Serialize the index object utilizing pickle.

p_cp = pickle.hundreds(pickle.dumps(hnsw_index))

Deleting parts.

for id in ids2:
    hnsw_index.mark_deleted(id)

This can liberate the final 100 parts from the index. If you want, you can too reuse the reminiscence of deleted parts.

hnsw_index.add_items(data3, labels3, replace_deleted=True)

Conclusion

The HNSW is without doubt one of the most vital algorithms proper now for the event of vector retrieval strategies. It’s the main indexing algorithm utilized in all main vector databases. Hope you’ve understood the workings of HNSW by this text. As AI evolves, we are going to see the event of bigger and extra advanced studying fashions, inflicting an increase within the want for utilizing HNSW and rising its functions and significance.

Key Takeaways

• Vector databases are purpose-built information shops for storing high-dimensional vector embeddings.
• Indexing of embeddings permits vector shops to deal with querying, insertion, and deletion of embeddings.
• There are alternative ways of indexing vectors, reminiscent of IVF, Annoy, Quantization, and HNSW.
• HNSW is a mixture of two algorithms. Skip lists and a Navigable Small World (NSW).

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