Oxford 05-oct-2012


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Algorithmic details of how to do on-line k-means clustering

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  • 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.
  • Oxford 05-oct-2012

    1. 1. Fast Single-pass k-means Clustering
    2. 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
    3. 3. Agenda• Rationale• Theory – clusterable data, k-mean failure modes, sketches• Algorithms – ball k-means, surrogate methods• Implementation – searchers, vectors, clusterers• Results• Application
    4. 4. RATIONALE
    5. 5. 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
    6. 6. 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
    7. 7. An Example
    8. 8. 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
    9. 9. An Example
    10. 10. An Example
    11. 11. 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
    12. 12. Diagonalized Cluster Proximity
    13. 13. Lots of Clusters Are Fine
    14. 14. 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
    15. 15. THEORY
    16. 16. Intuitive Theory• Traditionally, minimize over all distributions – optimization is NP-complete – that isn’t like real data• Recently, assume well-clusterable data s 2 D 2 (X) > D 2 (X) k-1 k• Interesting approximation bounds provable 1+ O(s 2 )
    17. 17. For Example 1 D (X) > 2 D (X) 2 s 4 2 5 Grouping these two clusters seriously hurts squared distance
    18. 18. ALGORITHMS
    19. 19. 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
    20. 20. Typical k-means Failure Selecting two seeds here cannot be fixed with Lloyds Result is that these two clusters get glued together
    21. 21. 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
    22. 22. 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
    23. 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. 24. 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
    25. 25. 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 clusters may suffice
    26. 26. 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 – log k + log log n = 11 + 5 = 17 – 30,000 times faster is a bona fide big deal
    27. 27. 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
    28. 28. 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• 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
    30. 30. 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
    31. 31. Projection Search total ordering!
    32. 32. How Many Projections?
    33. 33. 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
    34. 34. 1 LSH Bit-match Versus Cosine 0.8 0.6 0.4 0.2Y Ax is 0 0 8 16 24 32 40 48 56 64 - 0.2 - 0.4 - 0.6 - 0.8 -1 X Ax is
    35. 35. Results with 32 Bits
    36. 36. The Internals• Mechanism for extending Mahout Vectors – DelegatingVector, WeightedVector, Centroid• Searcher interface – ProjectionSearch, KmeansSearch, LshSearch, Brute• Super-fast clustering – Kmeans, StreamingKmeans
    37. 37. Parallel Speedup? 200 Non- threaded ✓ 100 2Tim 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
    38. 38. 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
    39. 39. 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
    40. 40. APPLICATION
    41. 41. The Business Case• Our customer has 100 million cards in circulation• Quick and accurate decision-making is key. – Marketing offers – Fraud prevention
    42. 42. Opportunity• Demand of modeling is increasing rapidly• So they are testing something simpler and more agile• Like k-nearest neighbor
    43. 43. 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
    44. 44. K-Nearest Neighbor Example
    45. 45. 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
    46. 46. 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!
    47. 47. 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
    48. 48. Lots of Clusters Are Fine
    49. 49. Lots of Clusters Are Fine
    50. 50. 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
    51. 51. Summary• Nearest neighbor algorithms can be blazing fast• But you need blazing fast clustering – Which we now have
    52. 52. 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/speaking/strata-10- 2-12• Contact me at tdunning@maprtech.com or @ted_dunning