# Nearest Neighbor Descent

library(rnndescent)

Nearest Neighbor Descent (Dong, Moses, and Li 2011) (NND) is the main way to construct a k-nearest neighbors graph in rnndescent. Here’s a brief description of the method.

The idea behind NND is to start with an initial guess of the graph (typically randomly chosen neighbors) and then iteratively improving that guess by taking candidate neighbors which are neighbors of neighbors. For example: if an item i has a current neighbor j, then j’s neighbors are candidates for i’s neighbors. The “descent” part is in analogy with gradient descent, where you can see the sum of the distances in the graph as an objective function: as better neighbors enter the graph, the distances must get smaller.

Conceptually it would seem that you would implement this algorithm with a loop like the following in each iteration:

1. For each item i in the graph:
2. For each item j in the neighbors of i:
3. For each item k in the neighbors of j:
4. If k is not already a neighbor of i:
5. Calculate the distance between i and k, $$d_{ik}$$.
6. If $$d_{ik}$$ is smaller than the neighbor with the largest distance in the neighbor list of $$i$$, update the neighbor list of i with k.

## Local Join

The process described above involves a lot of looping and repeated fetching of neighbor vectors, so NND actually uses the concept of a “local join”. One way to think of it is to consider an item i fielding requests for its nearest neighbors. It will be repeatedly asked for it by any other item which considers it a neighbor. So if we did some work at the start of each iteration to know all the items which consider i a neighbor, we can generate all the candidates neighbor pairs that i is involved with at once. Then we only need to iterate over the items in the graph. We do need to do the work of finding out who considers i a neighbor but that also only requires a loop over the graph also.

To be clear, the same amount of work needs to be done, but by doing it in a different order, everything is a bit more efficient in terms of what needs to be fetched from memory.

The up-shot of using the local join approach is that rather than iterating over the graph one item at a time, we end up a list of pairs of items (i, j) to update the graph as a whole with. And because we are dealing with a kNN graph if we have a pair (i, j) we also have (j, i) as a potential update, at the cost of only one distance calculation. This has some challenges in terms of parallel implementation and it also makes caching distances a bit harder but it’s still better than the more naive approach of explicitly looping over all neighbors-of-neighbors.

## Other Heuristics

Additionally, there are two other heuristics used to reduce the amount of work done. The first is that candidate neighbors are split into “new” and “old” candidates. A “new” candidate neighbor is any neighbor which entered the graph in the previous iteration. “Old” neighbors are everything else. For the local join, all possible pairs of “new” neighbors are used for updating the graph, but “old” neighbors are only ever paired with “new” neighbors, not other “old” neighbors. This is referred to as “incremental search” in the NND paper.

Also, a tolerance $$\delta$$ is used to determine as an early stopping criterion. The total number of items in the graph is $$kN$$ where $$k$$ is the number of neighbors and $$N$$ is the number of items. During each iteration, a counter is incremented every time the graph is successfully updated. If at the end of the iteration the number of updates is less than $$\delta kN$$ then the iteration stops.

## PyNNDescent Modifications

There is one other minor change to how PyNNDescent works versus the description in the NND paper, which rnndescent also uses, which is how sampling of candidates works. For the local join, we need to know not just the neighbors of i, but those items which consider i a neighbor, which we call the “reverse neighbors” of i. While there are always only $$k$$ “forward” neighbors of i in a graph, we don’t control who is a neighbor of what, so i could be the neighbor of many (or even all) the other items in a dataset. Thus, building the reverse list can be a bit challenging as we need to be prepared for any item to have up to $$N$$ neighbors. In the NND paper, this is avoided by defining a sample rate $$\rho$$, which is used to sample from the k-nearest neighbors, and then the reverse neighbor list is only built from the sampled items. A subsequent down-sampling is then applied to the reverse neighbor list so that both the forward and reverse neighbor list only contain $$\rho k$$ items.

Instead of a sample rate, rnndescent defines a max_candidates parameter determines the size of both the forward and reverse neighbor lists per item. If there are more candidates than the max_candidates value, the retained candidates are chosen randomly so this works like random sampling.

Finally, instead of a random initialization, PyNNDescent uses a k-nearest neighbors graph from a random projection forest. There is an entire vignette explaining how RP forest works. This is also an option in rnndescent.

## Example

It’s easy enough to run NND on a dataset. Here’s an example using the iris dataset:

iris_knn <- nnd_knn(iris, k = 15)

The contents of iris_knn is a list with two elements, both $$N$$ by $$k$$ matrices where $$N$$ is the number of items in the dataset and $$k$$ is the number of neighbors: idx contains the indices of the neighbors:

iris_knn$idx[1:2, 1:5] #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1 18 5 29 40 #> [2,] 2 13 46 35 10 and dist contains the distances: iris_knn$dist[1:2, 1:5]
#>      [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,]    0 0.1000000 0.1414212 0.1414212 0.1414213
#> [2,]    0 0.1414213 0.1414213 0.1414213 0.1732050

Apart from k, there are some parameters you may want to modify:

• metric is the distance metric to use. The default is Euclidean distance. There are several metrics you can use. See the documentation for nnd_knn for the full list.
• init is the initialization method. The default is "rand" which initializes the neighbors randomly. You may wish to use "tree" which uses a random projection forest to initialize the neighbors, similar to rpf_build. To control the tree building, you can pass the same sort of parameters that you would to rpf_build via the init_args parameter. See the vignette on RP forest for more details. You can also pass in a neighbor graph directly. This should have the same format as the output of nnd_knn, i.e. a list of two matrices of size $$N$$ by $$k$$. NND can be used to refine an existing graph generated by other methods, e.g. RcppAnnoy or RcppHNSW.
• n_iters is the number of iterations of NND to carry out. The default is to choose based on $$N$$, the number of items in the dataset. The amount of work done per iteration decreases quite rapidly, so sticking with the default is usually sensible, especially if you don’t change the convergence criterion delta (see below), because this often causes the algorithm to stop early anyway.
• delta controls early stopping and must be a value between 0 and 1. If in a given iteration, the number of changes to the neighbor graph is less than delta * k * N then the algorithm stops. The default is 0.001 so you can interpret that roughly as the neighbor graph needs to have changed by 0.1% to avoid early stopping.
• max_candidates controls the size of the forward and reverse neighbor lists. The default is to set this to whatever is smaller, k or 60.
• n_threads controls the number of threads to use. The default is to run as a single thread. The slow part of any approximate nearest neighbor algorithm is the distance calculation so using multiple threads is usually a good idea.
• ret_forest if TRUE, and you have set init = "tree", then the random projection forest used to initialize the neighbor graph is returned as well. If you want to generate new neighbors based on the original data you will want this.
• verbose set to TRUE to get information about the progress of the NND.
• progress this affects how the progress of NND is displayed when verbose = TRUE. The default bar shows a textual progress bar. You can also set progress = "dist" to show the current value of the convergence criterion and the sum of the distances at each iteration. This can help a bit to determine if more iterations or a different convergence criterion will help.

Note that NND uses random number generation to determine the order of processing candidates, so for reproducible results you should set the random number seed explicitly. Also, the way that parallelism is implemented means that reproducibility is not possible for different settings of n_threads even with a consistent seed, e.g. going from n_threads = 0 to n_threads = 4 will give you different results, even if you set.seed with a fixed seed beforehand.

## Troubleshooting

If you have reason to believe you aren’t getting the results out of NND that are sufficiently accurate, probably the best thing to do is to increase max_candidates. Reducing delta or increasing n_iters usually has less effect. Restarting nnd_knn with init set to the output of your previous run usually also helps, but is not a very time-efficient way to improve matters.

Here is some (lightly edited) sample output when running

iris_knn <- nnd_knn(iris, k = 15, verbose = TRUE, progress = "dist")
Running nearest neighbor descent for 7 iterations
1 / 7
heap sum = 647.85 num_updates = 3356 tol = 2.25
2 / 7
heap sum = 599.9 num_updates = 216 tol = 2.25
3 / 7
heap sum = 599.9 num_updates = 0 tol = 2.25
Convergence: c = 0 tol = 2.25

This tells you that for a dataset of the size of iris, at most 7 iterations will run. The 1 / 7, 2 / 7 and so on is logged at the end of each iteration. Following that is the sum of the distances of the neighbors in the heap, the number of updates to the neighbor graph and the convergence criterion. If num_updates falls below tol then the algorithm stops. In this case, on the third iteration there were no updates at all, so the algorithm stopped early.

In this case, almost certainly NND has found the exact nearest neighbors, so you wouldn’t be worried about modifying the parameters. But if you were so inclined, the output shows you that there would be little point in increasing n_iters or reducing delta. This really only leaves max_candidates as an option.

The vignette on dealing with hubness (where this can be an issue) goes into a bit more detail on how to use different functions in rnndescent to deal with this sort of problem.

## Querying New Data

You can’t. NND can only produce the k-nearest neighbors graph for the provided data. It doesn’t produce an “index” of any kind that you can query. The value of NND and the local join really only makes sense if you can take advantage of the fact that calculating the distance $$d_{ij}$$ lets update the neighbor list of $$i$$ and $$j$$ at once.

If you try to apply the concepts from NND to querying new data you quickly end up at a method that looks a lot like most greedy graph-based searches. For that, you should look at graph_knn_query(), although as noted above you can also use the random projection forest used to initialize the neighbor graph when init = "tree". You will also probably want to augment the neighbor graph to make it more amenable for searching using prepare_search_graph().

## References

Dong, Wei, Charikar Moses, and Kai Li. 2011. “Efficient k-Nearest Neighbor Graph Construction for Generic Similarity Measures.” In Proceedings of the 20th International Conference on World Wide Web, 577–86.