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
This slide is red to indicate missing data
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.
Transcript
Fast Single-pass k-means
Clustering
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
Why k-means?
• Clustering allows fast search
– k-nn models allow agile modeling
– lots of data points, 108 typical
– lots of clusters, 104 typical
• Model features
– Distance to nearest centroids
– Poor man’s manifold discovery
What is Quality?
• Robust clustering not a goal
– we don’t care if the same clustering is replicated
• Generalization to unseen data critical
– number of points per cluster
– distance distributions
– target function distributions
– model performance stability
• Agreement to “gold standard” is a non-issue
An Example
The Problem
• Spirals are a classic “counter” example for k-
means
• Classic low dimensional manifold with added
noise
• But clustering still makes modeling work well
An Example
An Example
The Cluster Proximity Features
• Every point can be described by the nearest
cluster
– 4.3 bits per point in this case
– Significant error that can be decreased (to a point)
by increasing number of clusters
• Or by the proximity to the 2 nearest clusters (2
x 4.3 bits + 1 sign bit + 2 proximities)
– Error is negligible
– Unwinds the data into a simple representation
Diagonalized Cluster Proximity
Lots of Clusters Are Fine
The Limiting Case
• Too many clusters lead to over-fitting
• Which we mediate by averaging over several
nearby clusters
• In the limit we get k-nn modeling
– and probably use k-means to speed up search
THEORY
Intuitive Theory
• Traditionally, minimize over all distributions
– optimization is NP-complete
– that isn’t like real data
• Recently, assume well-clusterable data
• Interesting approximation bounds provable
s 2
Dk-1
2
(X) > Dk
2
(X)
1+O(s 2
)
For Example
Grouping these
two clusters
seriously hurts
squared distance
D4
2
(X) >
1
s 2
D5
2
(X)
ALGORITHMS
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
Typical k-means Failure
Selecting two seeds
here cannot be
fixed with Lloyds
Result is that these two
clusters get glued
together
Ball k-means
• Provably better for highly clusterable data
• Tries to find initial centroids in each “core” of each 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
Still Not a Win
• Ball k-means is nearly guaranteed with k = 2
• Probability of successful seeding drops
exponentially with k
• Alternative strategy has high probability of
success, but takes O(nkd + k3d) time
Not good enough
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
Algorithm Costs
• O(k d log n) per point per iteration for Lloyd’s
algorithm
• Number of iterations not well known
• Iteration > log n reasonable assumption
Algorithm Costs
• 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
O(κ k d + k3 d) = O(k2 d log n + k3 d) for small k, high quality
O(κ d log k) or O(d log κ log k) for larger k, looser quality
– result is k high-quality centroids
• Even the sloppy surrogate may suffice
Algorithm Costs
• How much faster for the sketch phase?
– take k = 2000, d = 10, n = 100,000
– k d log n = 2000 x 10 x 26 = 500,000
– d (log k + log log n) = 10(11 + 5) = 170
– 3,000 times faster is a bona fide big deal
Pragmatics
• But this requires a fast search internally
• Have to cluster on the fly for sketch
• Have to guarantee sketch quality
• Previous methods had very high complexity
How It Works
• For each point
– Find approximately nearest centroid (distance = d)
– If (d > threshold) new centroid
– Else if (u > d/threshold) new cluster
– Else add to nearest centroid
• If centroids > κ ≈ C log N
– Recursively cluster centroids with higher threshold
Resulting Surrogate
• 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
• Either way, we win
IMPLEMENTATION
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
Projection Search
total ordering!
How Many Projections?
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
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 shows same linear
speedup
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
– Ball k-means strategy 1 performance is very good
with large k
How Well Does it Work?
• Empirical evaluation on 20 newsgroups
• Alternative algorithms include ball k-means
versus streaming k-means|ball k-means
• Results
Average distance to nearest cluster on held-out data
same or slightly smaller
Median distance to nearest cluster is smaller
> 10x faster (I/O and encoding limited)
APPLICATION
The Business Case
• Our customer has 100 million cards in
circulation
• Quick and accurate decision-making is key.
– Marketing offers
– Fraud prevention
Opportunity
• Demand of modeling is increasing rapidly
• So they are testing something simpler and
more agile
• Like k-nearest neighbor
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
K-Nearest Neighbor Example
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
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!
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
Lots of Clusters Are Fine
Lots of Clusters Are Fine
Some Details
• Clumpy data works better
– Real data is clumpy
• Speedups of 100-200x seem practical with
50% overlap
– Projection search and LSH give additional 100x
• More experiments needed
Summary
• Nearest neighbor algorithms can be blazing
fast
• But you need blazing fast clustering
– Which we now have
Contact Me!
• We’re hiring at MapR in US and Europe
• MapR software available for research use
• Come get the slides at
http://www.mapr.com/company/events/acmsf-2-25-13
• Get the code as part of Mahout trunk
• Contact me at tdunning@maprtech.com or @ted_dunning
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