Recent algorithmic developments [1] have enabled dramatic improvements in performance for clustering applications. Previously, the workhorse clustering algorithm was k-means which scaled linearly with the desired number of clusters times the data size times the number of iterations required. The number of iterations itself depended on the number of clusters and in map-reduce implementations such as in Mahout [2], the required iterative implementation is exceedingly painful. These new algorithms require only a single pass over the data and each pass has a cost that is roughly O(log k) where k is the desired number of clusters. The resulting implementation [3] which is being ported into Mahout has demonstrated some stunning speed. In one test, a uni-processor threaded implementation demonstrated the ability to cluster data points in just 20 micro-seconds per data point. Moreover, this algorithm is easily ported to map-reduce with essentially perfect linear scaling. This implies we should be able to cluster hundreds of millions of data points in minutes on moderate sized cluster. Even more exciting, these algorithms are online algorithms, so it is possible to build a real-time clustering engine that clusters data points as they arrive and never needs to look back at old data. I will talk about the basic intuitions behind these algorithms, how they are implemented, their limitations and how to use them. I will also talk about some of the very exciting practical implications of having a super-fast clustering algorithm.

1. Super-fast Online Clustering
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2. whoami – Ted Dunning
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3. Clustering? Why?
 Because other people do it
– Really!
 Because cluster distances make great model features
– Better
 Because good clusters help with really fast nearest neighbor search
– Very nice
 Because we can use clusters as a surrogate for all the data
– And that lets us train models or do visualization
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4. Agenda
 Nearest neighbor models
– Colored dots; need good distance metric; projection, LSH and k-means
search
 K-means algorithms
– O(k d log n) per point for Lloyd’s algorithm
… not good for k = 2000, n = 108
– Surrogate methods
• fast, sloppy single pass clustering with κ = k log n
• fast sloppy search for nearest cluster, O(log κ) = O(d (log k + log log n)) per point
• fast, in-memory, high-quality clustering of κ weighted centroids
• result consists of k high-quality centroids for the original data
 Results
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5. Nearest Neighbor Models
 Find the k nearest training examples
 Use the average value of the target variable from them
 This is easy … but hard
– easy because it is so conceptually simple and you don’t have knobs to turn
or models to build
– hard because of the stunning amount of math
– also hard because we need top 50,000 results
 Initial rapid prototype was massively too slow
– 3K queries x 200K examples takes hours
– needed 20M x 25M in the same time
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6. K-Nearest Neighbor Example
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7. Comparison to Other Modeling Approaches
 Logistic regression
– Depends on linear separability
– k-nn works very well if logistic regression works
– k-nn can work very well even if logistic regression fails due to interactions
producing non-linear decision surface
 Tree-based methods
– mostly roughly equivalent in accuracy with k-nn
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8. Required Scale and Speed and Accuracy
 Want 20 million queries against 25 million references in 10,000 s
 Should be able to search > 100 million references
 Should be linearly and horizontally scalable
 Must have >50% overlap against reference search
 Evaluation by sub-sampling is viable, but tricky
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9. How Hard is That?
 20 M x 25 M x 100 Flop = 50 P Flop
 1 CPU = 5 Gflops
 We need 10 M CPU seconds => 10,000 CPU’s
 Real-world efficiency losses may increase that by 10x
 Not good!
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10. How Can We Search Faster?
 First rule: don’t do it
– If we can eliminate most candidates, we can do less work
– Projection search and k-means search
 Second rule: don’t do it
– We can convert big floating point math to clever bit-wise integer math
– Locality sensitive hashing
 Third rule: reduce dimensionality
– Projection search
– Random projection for very high dimension
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11. Note the Circularity
 Clustering helps nearest neighbor search
 But clustering needs nearest neighbor search internally
 How droll !
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12. Projection Search
java.lang.TreeSet!
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13. How Many Projections?
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14. LSH Search
 Each random projection produces independent sign bit
 If two vectors have the same projected sign bits, they probably
point in the same direction (i.e. cos θ ≈ 1)
 Distance in L2 is closely related to cosine
x - y 2 = x - 2(x × y) + y
2 2
= x 2 - 2 x y cosq + y 2
 We can replace (some) vector dot products with long integer XOR
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15. LSH Bit-match Versus Cosine
1
0.8
0.6
0.4
0.2
Y Ax is
0
0 8 16 24 32 40 48 56 64
- 0.2
- 0.4
- 0.6
- 0.8
-1
X Ax is
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16. Results
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17. K-means Search
 First do clustering with lots (thousands) of clusters
 Then search nearest clusters to find nearest points
 We win if we find >50% overlap with “true” answer
 We lose if we can’t cluster super-fast
– more on this later
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18. Lots of Clusters Are Fine
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19. Lots of Clusters Are Fine
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20. Some Details
 Clumpy data works better
– Real data is clumpy 
 Speedups of 100-200x seem practical with 50% overlap
– Projection search and LSH can be used to accelerate that (some)
 More experiments needed
 Definitely need fast search
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21. So Now Some Clustering
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22. Lloyd’s Algorithm
 Part of CS folk-lore
 Developed in the late 50’s for signal quantization, published in 80’s
initialize k cluster centroids somehow
for each of many iterations:
for each data point:
assign point to nearest cluster
recompute cluster centroids from points assigned to clusters
 Highly variable quality, several restarts recommended
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23. Ball k-means
 Provably better for highly clusterable data
 Tries to find initial centroids in the “core” of real clusters
 Avoids outliers in centroid computation
initialize centroids randomly with distance maximizing tendency
for each of a very few iterations:
for each data point:
assign point to nearest cluster
recompute centroids using only points much closer than closest cluster
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24. Surrogate Method
 Start with sloppy clustering into κ = k log n clusters
 Use these clusters as a weighted surrogate for the data
 Cluster surrogate data using ball k-means
 Results are provably high quality for highly clusterable data
 Sloppy clustering can be done on-line
 Surrogate can be kept in memory
 Ball k-means pass can be done at any time
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25. Algorithm Costs
 O(k d log n) per point for Lloyd’s algorithm
… not so good for k = 2000, n = 108
 Surrogate methods
– fast, sloppy single pass clustering with κ = k log n
– fast sloppy search for nearest cluster, O(d log κ) = O(d (log k + log log n))
per point
– fast, in-memory, high-quality clustering of κ weighted centroids
– result consists of k high-quality centroids
 This is a big deal:
– k d log n = 2000 x 10 x 26 = 50,000
– log k + log log n = 11 + 5 = 17
– 3000 times faster makes the grade as a bona fide big deal
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26. The Internals
 Mechanism for extending Mahout Vectors
– DelegatingVector, WeightedVector, Centroid
 Searcher interface
– ProjectionSearch, KmeansSearch, LshSearch, Brute
 Super-fast clustering
– Kmeans, StreamingKmeans
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27. How It Works
 For each point
– Find approximately nearest centroid (distance = d)
– If d > threshold, new centroid
– Else possibly new cluster
– Else add to nearest centroid
 If centroids > K ~ C log N
– Recursively cluster centroids with higher threshold
 Result is large set of centroids
– these provide approximation of original distribution
– we can cluster centroids to get a close approximation of clustering original
– or we can just use the result directly
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28. Parallel Speedup?
200
Non- threaded
✓
100
2
Tim e per point (μs)
Threaded version
3
50
4
40 6
5
8
30
10 14
12
20 Perfect Scaling 16
10
1 2 3 4 5 20
Threads
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29. What About Map-Reduce
 Map-reduce implementation is nearly trivial
– Compute surrogate on each split
– Total surrogate is union of all partial surrogates
– Do in-memory clustering on total surrogate
 Threaded version shows linear speedup already
– Map-reduce speedup is likely, not entirely guaranteed
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30. How Well Does it Work?
 Theoretical guarantees for well clusterable data
– Shindler, Wong and Meyerson, NIPS, 2011
 Evaluation on held-out data
– Need results here
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31. Summary
 Nearest neighbor algorithms can be blazing fast
 But you need blazing fast clustering
– Which we now have
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32. Contact Us!
 We’re hiring at MapR in California
 Contact Ted at tdunning@maprtech.com or @ted_dunning
 For slides and other info
http://www.mapr.com/company/events/speaking/la-hug-9-25-12
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