The document discusses new techniques for improving the k-means clustering algorithm. It begins by describing the standard k-means algorithm and Lloyd's method. It then discusses issues with random initialization for k-means. It proposes using furthest point initialization (k-means++) as an improvement. The document also discusses parallelizing k-means initialization (k-means||) and using nearest neighbor data structures to speed up assigning points to clusters, which allows k-means to scale to many clusters. Experimental results show these techniques provide faster and higher quality clustering compared to standard k-means.

1. Teaching k-Means New Tricks
Sergei Vassilvitskii
Google

2. k-Means Algorithm
The k-Means Algorithm [Lloyd ’57]
– Clusters points intro groups
– Remains a workhorse of machine learning even in the age of deep networks

3. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Initialize with random clusters
49

4. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Assign each point to nearest center
50

5. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Recompute optimum centers (means)
51

6. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat: Assign points to nearest center
52

7. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat: Recompute centers
53

8. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat...
54

9. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat...Until clustering does not change
55

10. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat...Until clustering does not change
Total error reduced at every step - guaranteed to converge.
55

11. MR ML Algorithmics Sergei Vassilvitskii
Lloyd’s Method: k-means
Repeat...Until clustering does not change
Total error reduced at every step - guaranteed to converge.
Minimizes:
56
(X, C) =
X
x2X
d(x, C)2

12. New Tricks for k-Means
Initialization:
– Is random initialization a good idea?
Large data:
– Clustering many points (in parallel)
– Clustering into many clusters

13. MR ML Algorithmics Sergei Vassilvitskii
k-means Initialization
Random?
57

14. MR ML Algorithmics Sergei Vassilvitskii
k-means Initialization
Random?
58

15. MR ML Algorithmics Sergei Vassilvitskii
k-means Initialization
Random? A bad idea
59

16. MR ML Algorithmics Sergei Vassilvitskii
k-means Initialization
Random? A bad idea
Even with many random restarts!
59

17. MR ML Algorithmics Sergei Vassilvitskii
Easy Fix
Select centers using a furthest point algorithm (2-approximation to k-
Center clustering).
60

18. MR ML Algorithmics Sergei Vassilvitskii
Easy Fix
Select centers using a furthest point algorithm (2-approximation to k-
Center clustering).
61

19. MR ML Algorithmics Sergei Vassilvitskii
Easy Fix
Select centers using a furthest point algorithm (2-approximation to k-
Center clustering).
62

20. MR ML Algorithmics Sergei Vassilvitskii
Easy Fix
Select centers using a furthest point algorithm (2-approximation to k-
Center clustering).
63

21. MR ML Algorithmics Sergei Vassilvitskii
Easy Fix
Select centers using a furthest point algorithm (2-approximation to k-
Center clustering).
64

22. MR ML Algorithmics Sergei Vassilvitskii
Sensitive to Outliers
65

23. MR ML Algorithmics Sergei Vassilvitskii
Sensitive to Outliers
65

24. MR ML Algorithmics Sergei Vassilvitskii
Sensitive to Outliers
65

25. MR ML Algorithmics Sergei Vassilvitskii
Sensitive to Outliers
65

26. MR ML Algorithmics Sergei Vassilvitskii
Sensitive to Outliers
66

27. MR ML Algorithmics Sergei Vassilvitskii
Interpolate between two methods. Give preference to further points.
Let be the distance between and the nearest cluster center.
Sample next center proportionally to .
k-means++
67
D(p) p
D↵
(p)

28. MR ML Algorithmics Sergei Vassilvitskii
k-means++
68
D(p) p
Interpolate between two methods. Give preference to further points.
Let be the distance between and the nearest cluster center.
Sample next center proportionally to .D↵
(p)
D↵
(p)
P
x D↵(p)
kmeans++:
Select first point uniformly at random
for (int i=1; i < k; ++i){
Select next point p with probability ;
UpdateDistances();
}

29. MR ML Algorithmics Sergei Vassilvitskii
k-means++
69
D(p) p
Interpolate between two methods. Give preference to further points.
Let be the distance between and the nearest cluster center.
Sample next center proportionally to .D↵
(p)
↵ = 1
↵ = 2
Original Lloyd’s:
Furthest Point:
k-means++:
↵ = 0
D↵
(p)
P
x D↵(p)
kmeans++:
Select first point uniformly at random
for (int i=1; i < k; ++i){
Select next point p with probability ;
UpdateDistances();
}

30. MR ML Algorithmics Sergei Vassilvitskii
k-means++
70

31. MR ML Algorithmics Sergei Vassilvitskii
k-means++
70

32. MR ML Algorithmics Sergei Vassilvitskii
k-means++
70

33. MR ML Algorithmics Sergei Vassilvitskii
k-means++
70

34. MR ML Algorithmics Sergei Vassilvitskii
k-means++
71
Theorem [AV ’07]: k-means++ guarantees a approximation⇥(log k)

35. New Tricks for k-Means
Initialization:
– Is random initialization a good idea?
Large data:
– Clustering many points (in parallel)
– Clustering into many clusters

36. Dealing with large data
The new initialization approach:
– Leads to very good clusterings
– But is very sequential!
• Must select one cluster at a time, then update the distribution we are
sampling from
– How to adapt it in the world of parallel computing?

37. Speeding up initialization
Initialization:
kmeans++:
Select first point uniformly at random
for (int i=1; i < k; ++i) {
Select next point p with probability ;
UpdateDistance();
}
Improving the speed:
– Instead of selecting a single point, sample many points at a time
– Oversample: select more than k centers, and then select the best k out of them.
D2
(p)
P
x D2(x)

38. MR ML Algorithmics Sergei Vassilvitskii
k-means||
74
kmeans++:
Select first point uniformly at random
for (int i=1; i < k; ++i){
Select next point p with probability ;
UpdateDistances();
}
}
D2
(p)
P
p D2(p)

39. MR ML Algorithmics Sergei Vassilvitskii
k-means||
75
kmeans++:
Select first point c uniformly at random
for (int i=1; i < ; ++i){
Select point p independently with probability
UpdateDistances();
}
Prune to k points total by clustering the clusters
}
k · ` ·
D↵
(p)
P
x D↵(p)
log`( (X, c))

40. MR ML Algorithmics Sergei Vassilvitskii
k-means||
76
kmeans++:
Select first point c uniformly at random
for (int i=1; i < ; ++i){
Select point p independently with probability
UpdateDistances();
}
Prune to k points total by clustering the clusters
}
k · ` ·
D↵
(p)
P
x D↵(p)
log`( (X, c))
Independent selection
Easy MR

41. MR ML Algorithmics Sergei Vassilvitskii
k-means||
77
kmeans++:
Select first point c uniformly at random
for (int i=1; i < ; ++i){
Select point p independently with probability
UpdateDistances();
}
Prune to k points total by clustering the clusters
}
k · ` ·
D↵
(p)
P
x D↵(p)
log`( (X, c))
Independent selection
Easy MR
Oversampling Parameter

42. MR ML Algorithmics Sergei Vassilvitskii
k-means||
78
kmeans++:
Select first point c uniformly at random
for (int i=1; i < ; ++i){
Select point p independently with probability
UpdateDistances();
}
Prune to k points total by clustering the clusters
}
k · ` ·
D↵
(p)
P
x D↵(p)
log`( (X, c))
Independent selection
Easy MR
Oversampling Parameter
Re-clustering step

43. MR ML Algorithmics Sergei Vassilvitskii
k-means||: Analysis
How Many Rounds?
– Theorem: After rounds, guarantee approximation
– In practice: fewer iterations are needed
– Need to re-cluster intermediate centers
Discussion:
– Number of rounds independent of k
– Tradeoff between number of rounds and memory
79
O(1)O(log`(n ))
O(k` log`(n ))

44. MR ML Algorithmics Sergei Vassilvitskii
How well does this work?
80
1e+12
1e+13
1 10
log # Rounds
1e+11
1e+12
1e+13
1
1e+11
1e+12
1e+13
1e+14
1e+15
1e+16
1 10
cost
log # Rounds
KDD Dataset, k=65
l/k=1
l/k=2
l/k=4
1e+10
1e+11
1e+12
1e+13
1e+14
1e+15
1e+16
1
cost
Random Initialization
k-means++
k-means||
l=1
l=2
l=4

45. MR ML Algorithmics Sergei Vassilvitskii
Performance vs. k-means++
– Even better on small datasets: 4600 points, 50 dimensions (SPAM)
– Accuracy:
– Time (iterations):
81

46. New Tricks for k-Means
Initialization:
– Is random initialization a good idea?
Large data:
– Clustering many points (in parallel)
– Clustering into many clusters

47. Large k
How do you run k-means when k is large?
– For every point, need to find the nearest center

48. Large k
How do you run k-means when k is large?
– For every point, need to find the nearest center
– Naive approach: linear scan

49. Large k
How do you run k-means when k is large?
– For every point, need to find the nearest center
– Naive approach: linear scan
– Better approach [Elkan]:
• Use triangle inequality to see if the center could have possibly gotten closer
• Still expensive when k is large

50. Using Nearest Neighbor Data Structures
Expensive step of k-Means:
– For every point, find the nearest center
But we have many algorithms for nearest neighbors!

51. Using Nearest Neighbor Data Structures
Expensive step of k-Means:
– For every point, find the nearest center
But we have many algorithms for nearest neighbors!
First idea:
– Index the centers. Then do a query into this data structure for every point
– Need to rebuild the NN Data structure every time

52. Using Nearest Neighbor Data Structures
Expensive step of k-Means:
– For every point, find the nearest center
But we have many algorithms for nearest neighbors!
First idea:
– Index the centers. Then do a query into this data structure for every point
– Need to rebuild the NN Data structure every time
Better idea:
– Index the points!
– For every center, query the nearest points

53. Performance
Two large datasets:
– 1M points in each
– 7-25M features in each (very high dimensionality)
– Clustering into k=1000 clusters.

54. Performance
Two large datasets:
– 1M points in each
– 7-25M features in each (very high dimensionality)
– Clustering into k=1000 clusters.
Index based k-means:
– Simple implementation: 2-7x faster than traditional k-means
– No degradation in quality (same objective function value)
– More complex implementation:
• An additional 8-50x speed improvement !

55. K-Means Algorithm
Almost 60 years on, still incredibly popular and useful approach
It has gotten better with age:
– Better initialization approaches that are fast and accurate
– Parallel implementations to handle large datasets
– New implementations that handle points in many dimensions and clustering into
many clusters
– New approaches for online clustering

56. K-Means Algorithm
Almost 60 years on, still incredibly popular and useful approach
It has gotten better with age:
– Better initialization approaches that are fast and accurate
– Parallel implementations to handle large datasets
– New implementations that handle points in many dimensions and clustering into
many clusters
– New approaches for online clustering
More work remains!
– Non spherical clusters
– Other metric spaces
– Dealing with outliers

57. Thank You.
Arthur, D., V., S. K-means++, the advantages of better seeding. SODA 2007.
Bahmani, B., Moseley, B., Vattani A., Kumar, R., V.,S. Scalable k-means++.
VLDB 2012.
Broder, A., Garcia, L., Josifovski, V., V.S., Venkatesan, S. Scalable k-means by
ranked retrieval. WSDM 2014.