StreamingKMeans algorithm

The StreamingKMeans algorithm is a variant of Algorithm 1 from Shindler et al and consists of two steps:

  1. Streaming step
  2. BallKMeans step.

The streaming step is a randomized algorithm that makes one pass through the data and produces as many centroids as it determines is optimal. This step can be viewed as a preparatory dimensionality reduction. If the size of the data stream is n and the expected number of clusters is k, the streaming step will produce roughly k*log(n) clusters that will be passed on to the BallKMeans step which will further reduce the number of clusters down to k. BallKMeans is a randomized Lloyd-type algorithm that has been studied in detail, see Ostrovsky et al.

Streaming step


Overview

The streaming step is a derivative of the streaming portion of Algorithm 1 in Shindler et al. The main difference between the two is that Algorithm 1 of Shindler et al assumes the knowledge of the size of the data stream and uses it to set a key parameter for the algorithm. More precisely, the initial distanceCutoff (defined below), which is denoted by f in Shindler et al, is set to 1/(k(1+log(n)). The distanceCutoff influences the number of clusters that the algorithm will produce. In contrast, Mahout implementation does not require the knowledge of the size of the data stream. Instead, it dynamically re-evaluates the parameters that depend on the size of the data stream at runtime as more and more data is processed. In particular, the parameter numClusters (defined below) changes its value as the data is processed.

Parameters

  • numClusters (int): Conceptually, numClusters represents the algorithm's guess at the optimal number of clusters it is shooting for. In particular, numClusters will increase at run time as more and more data is processed. Note that •numClusters• is not the number of clusters that the algorithm will produce. Also, numClusters should not be set to the final number of clusters that we expect to receive as the output of StreamingKMeans.
  • distanceCutoff (double): a parameter representing the value of the distance between a point and its closest centroid after which the new point will definitely be assigned to a new cluster. distanceCutoff can be thought of as an estimate of the variable f from Shindler et al. The default initial value for distanceCutoff is 1.0/numClusters and distanceCutoff grows as a geometric progression with common ratio beta (see below).
  • beta (double): a constant parameter that controls the growth of distanceCutoff. If the initial setting of distanceCutoff is d0, distanceCutoff will grow as the geometric progression with initial term d0 and common ratio beta. The default value for beta is 1.3.
  • clusterLogFactor (double): a constant parameter such that clusterLogFactor log(numProcessedPoints) is the runtime estimate of the number of clusters to be produced by the streaming step. If the final number of clusters (that we expect StreamingKMeans to output) is k, clusterLogFactor can be set to k.
  • clusterOvershoot (double): a constant multiplicative slack factor that slows down the collapsing of clusters. The default value is 2.

Algorithm

The algorithm processes the data one-by-one and makes only one pass through the data. The first point from the data stream will form the centroid of the first cluster (this designation may change as more points are processed). Suppose there are r clusters at one point and a new point p is being processed. The new point can either be added to one of the existing r clusters or become a new cluster. To decide:

  • let c be the closest cluster to point p
  • let d be the distance between c and p
  • if d > distanceCutoff, create a new cluster from p (p is too far away from the clusters to be part of any one of them)
  • else (d <= distanceCutoff), create a new cluster with probability d / distanceCutoff (the probability of creating a new cluster increases as d increases).

There will be either r or r+1 clusters after processing a new point.

As the number of clusters increases, it will go over the clusterOvershoot * numClusters limit (numClusters represents a recommendation for the number of clusters that the streaming step should aim for and clusterOvershoot is the slack). To decrease the number of clusters the existing clusters are treated as data points and are re-clustered (collapsed). This tends to make the number of clusters go down. If the number of clusters is still too high, distanceCutoff is increased.

BallKMeans step


Overview

The algorithm is a Lloyd-type algorithm that takes a set of weighted vectors and returns k centroids, see Ostrovsky et al for details. The algorithm has two stages:

  1. Seeding
  2. Ball k-means

The seeding stage is an initial guess of where the centroids should be. The initial guess is improved using the ball k-means stage.

Parameters

  • numClusters (int): the number k of centroids to return. The algorithm will return exactly this number of centroids.

  • maxNumIterations (int): After seeding, the iterative clustering procedure will be run at most maxNumIterations times. 1 or 2 iterations are recommended. Increasing beyond this will increase the accuracy of the result at the expense of runtime. Each successive iteration yields diminishing returns in lowering the cost.

  • trimFraction (double): Outliers are ignored when computing the center of mass for a cluster. For any datapoint x, let c be the nearest centroid. Let d be the minimum distance from c to another centroid. If the distance from x to c is greater than trimFraction * d, then x is considered an outlier during that iteration of ball k-means. The default is 9/10. In Ostrovsky et al, the authors use trimFraction = 1/3, but this does not mean that 1/3 is optimal in practice.

  • kMeansPlusPlusInit (boolean): If true, the seeding method is k-means++. If false, the seeding method is to select points uniformly at random. The default is true.

  • correctWeights (boolean): If correctWeights is true, outliers will be considered when calculating the weight of centroids. The default is true. Note that outliers are not considered when calculating the position of centroids.

  • testProbability (double): If testProbability is p (0 < p < 1), the data (of size n) is partitioned into a test set (of size p*n) and a training set (of size (1-p)*n). If 0, no test set is created (the entire data set is used for both training and testing). The default is 0.1 if numRuns > 1. If numRuns = 1, then no test set should be created (since it is only used to compare the cost between different runs).

  • numRuns (int): This is the number of runs to perform. The solution of lowest cost is returned. The default is 1 run.

Algorithm

The algorithm can be instructed to take multiple independent runs (using the numRuns parameter) and the algorithm will select the best solution (i.e., the one with the lowest cost). In practice, one run is sufficient to find a good solution.

Each run operates as follows: a seeding procedure is used to select k centroids, and then ball k-means is run iteratively to refine the solution.

The seeding procedure can be set to either 'uniformly at random' or 'k-means++' using kMeansPlusPlusInit boolean variable. Seeding with k-means++ involves more computation but offers better results in practice.

Each iteration of ball k-means runs as follows:

  1. Clusters are formed by assigning each datapoint to the nearest centroid
  2. The centers of mass of the trimmed clusters (see trimFraction parameter above) become the new centroids

The data may be partitioned into a test set and a training set (see testProbability). The seeding procedure and ball k-means run on the training set. The cost is computed on the test set.

Usage of StreamingKMeans

 bin/mahout streamingkmeans  
   -i <input>  
   -o <output> 
   -ow  
   -k <k>  
   -km <estimatedNumMapClusters>  
   -e <estimatedDistanceCutoff>  
   -mi <maxNumIterations>  
   -tf <trimFraction>  
   -ri                  
   -iw  
   -testp <testProbability>  
   -nbkm <numBallKMeansRuns>  
   -dm <distanceMeasure>   
   -sc <searcherClass>  
   -np <numProjections>  
   -s <searchSize>   
   -rskm  
   -xm <method>  
   -h   
   --tempDir <tempDir>   
   --startPhase <startPhase>   
   --endPhase <endPhase>

Details on Job-Specific Options:

  • --input (-i) <input>: Path to job input directory.
  • --output (-o) <output>: The directory pathname for output.
  • --overwrite (-ow): If present, overwrite the output directory before running job.
  • --numClusters (-k) <k>: The k in k-Means. Approximately this many clusters will be generated.
  • --estimatedNumMapClusters (-km) <estimatedNumMapClusters>: The estimated number of clusters to use for the Map phase of the job when running StreamingKMeans. This should be around k * log(n), where k is the final number of clusters and n is the total number of data points to cluster.
  • --estimatedDistanceCutoff (-e) <estimatedDistanceCutoff>: The initial estimated distance cutoff between two points for forming new clusters. If no value is given, it's estimated from the data set
  • --maxNumIterations (-mi) <maxNumIterations>: The maximum number of iterations to run for the BallKMeans algorithm used by the reducer. If no value is given, defaults to 10.
  • --trimFraction (-tf) <trimFraction>: The 'ball' aspect of ball k-means means that only the closest points to the centroid will actually be used for updating. The fraction of the points to be used is those points whose distance to the center is within trimFraction * distance to the closest other center. If no value is given, defaults to 0.9.
  • --randomInit (-ri) Whether to use k-means++ initialization or random initialization of the seed centroids. Essentially, k-means++ provides better clusters, but takes longer, whereas random initialization takes less time, but produces worse clusters, and tends to fail more often and needs multiple runs to compare to k-means++. If set, uses the random initialization.
  • --ignoreWeights (-iw): Whether to correct the weights of the centroids after the clustering is done. The weights end up being wrong because of the trimFraction and possible train/test splits. In some cases, especially in a pipeline, having an accurate count of the weights is useful. If set, ignores the final weights.
  • --testProbability (-testp) <testProbability>: A double value between 0 and 1 that represents the percentage of points to be used for 'testing' different clustering runs in the final BallKMeans step. If no value is given, defaults to 0.1
  • --numBallKMeansRuns (-nbkm) <numBallKMeansRuns>: Number of BallKMeans runs to use at the end to try to cluster the points. If no value is given, defaults to 4
  • --distanceMeasure (-dm) <distanceMeasure>: The classname of the DistanceMeasure. Default is SquaredEuclidean.
  • --searcherClass (-sc) <searcherClass>: The type of searcher to be used when performing nearest neighbor searches. Defaults to ProjectionSearch.
  • --numProjections (-np) <numProjections>: The number of projections considered in estimating the distances between vectors. Only used when the distance measure requested is either ProjectionSearch or FastProjectionSearch. If no value is given, defaults to 3.
  • --searchSize (-s) <searchSize>: In more efficient searches (non BruteSearch), not all distances are calculated for determining the nearest neighbors. The number of elements whose distances from the query vector is actually computer is proportional to searchSize. If no value is given, defaults to 1.
  • --reduceStreamingKMeans (-rskm): There might be too many intermediate clusters from the mapper to fit into memory, so the reducer can run another pass of StreamingKMeans to collapse them down to a fewer clusters.
  • --method (-xm) method The execution method to use: sequential or mapreduce. Default is mapreduce.
  • -- help (-h): Print out help
  • --tempDir <tempDir>: Intermediate output directory.
  • --startPhase <startPhase> First phase to run.
  • --endPhase <endPhase> Last phase to run.

References

  1. M. Shindler, A. Wong, A. Meyerson: Fast and Accurate k-means For Large Datasets
  2. R. Ostrovsky, Y. Rabani, L. Schulman, Ch. Swamy: The Effectiveness of Lloyd-Type Methods for the k-means Problem