Ted Dunning describes an implementation of recent results that provide high quality k-means clustering at very high speed.
"For well clusterable data, this algorithm provides good bounds on quality, but practically speaking, it makes clustering practical in many applications by providing roughly 3 orders of magnitude speedup relative to the standard algorithm based on Lloyd's initial efforts. In addition, the algorithm is highly amenable to implementation using map-reduce and shows essentially linear speedup.
Just as significant, this new algorithm allows clustering with a very large number of clusters which makes it practical to use as a feature extraction algorithm or set up for a nearest neighbor search. " - Ted Dunning
2. whoami – Ted Dunning
• Chief Application Architect, MapR Technologies
• Committer, member, Apache Software Foundation
– particularly Mahout, Zookeeper and Drill
• Contact me at
tdunning@maprtech.com
tdunning@apache.com
ted.dunning@gmail.com
@ted_dunning
• Get slides and more info at
http://www.mapr.com/company/events/speaking/strata-10-2-12
3. Agenda
• Nearest neighbor models
• K-means algorithms
– O(k d log n) per point for Lloyd’s algorithm
– Surrogate (sketch) methods
• Results
5. The Business Case
• Our customer has 100 million cards in
circulation
• Quick and accurate decision-making is key.
– Marketing offers
– Fraud prevention
6. Opportunity
• Demand of modeling is increasing rapidly
• So we are testing something simpler and more
agile
• Like k-nearest neighbor
7. What’s that?
• Find the k nearest training examples – lookalike
customers
• 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
9. 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
10. 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!
11. 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
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
• We can replace (some) vector dot products with long
integer XOR
x - y 2
= x2
- 2(x× y)+ y2
= x2
- 2 x y cosq + y2
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
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
21. 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
22. 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
23. Surrogate Method
• Start with sloppy clustering into κ = k log n
clusters
• Use this sketch as a weighted surrogate for the
data
• Cluster surrogate data using ball k-means
• Results are provably good for highly clusterable
data
• Sloppy clustering is on-line
• Surrogate can be kept in memory
• Ball k-means pass can be done at any time
24. 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
26. 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
28. 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
29. How Well Does it Work?
• Theoretical guarantees for well clusterable
data
– Shindler, Wong and Meyerson, NIPS, 2011
• Evaluation on synthetic data
– Rough clustering produces correct surrogates
– Possible issue in ball k-means initialization (still
produces good clustering on test data)
30. Summary
• Nearest neighbor algorithms can be blazing
fast
• But you need blazing fast clustering
– Which we now have
31. Contact Us!
• We’re hiring at MapR in US and Europe
• Amex is hiring in Phoenix and New York
• Come get the slides at
http://www.mapr.com/company/events/speaking/strata-10-
2-12
• Contact Ted at tdunning@maprtech.com or
@ted_dunning
Editor's Notes
The basic idea here is that I have colored slides to be presented by you in blue. You should substitute and reword those slides as you like. In a few places, I imagined that we would have fast back and forth as in the introduction or final slide where we can each say we are hiring in turn.The overall thrust of the presentation is for you to make these points:Amex does lots of modelingit is expensivehaving a way to quickly test models and new variables would be awesomeso we worked on a new project with MapRMy part will say the following:Knn basic pictorial motivation (could move to you if you like)describe knn quality metric of overlapshow how bad metric breaks knn (optional)quick description of LSH and projection searchpicture of why k-means search is coolmotivate k-means speed as tool for k-means searchdescribe single pass k-means algorithmdescribe basic data structuresshow parallel speedupOur summary should state that we have achievedsuper-fast k-means clusteringinitial version of super-fast knn search with good overlap
The sub-bullets are just for reference and should be deleted later
The idea here is to guess what color a new dot should be by looking at the points within the circle. The first should obviously be purple. The second cyan. The third is uncertain, but probably isn’t green or cyan and probably is a bit more likely to be red than purple.