SwarmSVM

Tong He

1 Introduction

This package contains several ensemble learning algorithms based on the following papers:

  1. Gu, Q., & Han, J. (2013). Clustered support vector machines. In proceedings of the sixteenth international conference on artificial intelligence and statistics (pp. 307-315).
  2. Hsieh, C. J., Si, S., & Dhillon, I. S. (2013). A divide-and-conquer solver for kernel support vector machines. arXiv preprint arXiv:1311.0914.
  3. Collobert, R., Bengio, S., & Bengio, Y. (2002). A parallel mixture of SVMs for very large scale problems. Neural computation, 14(5), 1105-1114.

The main idea of these algorithms are

  1. Reducing the scale of the data set results in a faster algorithm.
  2. If we divide a linear inseperable problem into smaller sub-problems appropriately, then it is possible to solve it by linear SVMs.

These two ideas focus on the efficiency and accuracy respectively. Specifically, we usually use more than one SVM model to solve the whole problem, therefore this is also an ensemble learning framework.

2 Data

In this package, we choose a small data set to demonstrate the usage of our functions. The data set is svmguide1 from libsvm’s official website. The data is collected from an astroparticle application from Jan Conrad of Uppsala University, Sweden.

We can load it by

require(SwarmSVM)
data(svmguide1)

It is a list object. Let’s first take a look at the training data.

head(svmguide1[[1]])
##      [,1]       [,2]      [,3]          [,4]      [,5]
## [1,]    1 0.17830158 0.8459425 -0.0003471568 0.5025830
## [2,]    1 0.09974321 0.7418535  0.0013642697 0.6631010
## [3,]    1 0.12155597 0.8184516 -0.0006077044 0.5615699
## [4,]    1 0.18806891 0.6310490  0.0008937012 0.7525998
## [5,]    1 0.26460236 0.3487087 -0.0012692003 0.8991030
## [6,]    1 0.13390393 0.6548727 -0.0010820202 0.7437811

The first column contains the classification target value, the other columns contain the features. It is a binary classification task. The second part in the list is the test set:

head(svmguide1[[2]])
##      [,1]       [,2]      [,3]         [,4]      [,5]
## [1,]    0 0.04233811 0.2196920 -0.003501740 0.9746439
## [2,]    0 0.04233811 0.2196920 -0.003501740 0.9746439
## [3,]    0 0.01564010 0.7613563  0.002006680 0.6481419
## [4,]    0 0.03336976 0.2660178  0.004064065 0.9633818
## [5,]    0 0.02846241 0.3031386  0.003870255 0.9525135
## [6,]    0 0.01345867 0.4843888 -0.002009264 0.8747470

We rename them with the following command:

svmguide1.t = svmguide1[[2]]
svmguide1 = svmguide1[[1]]

From now on, we have the training data set svmguide1 and the test data set svmguide1.t.

3 Clustering Algorithm

In our pacakge, there are two main algorithms requiring clustering algorithm for the input data. Therefore we provide some clustering algorithms for users to choose from. Users can also implement their own functions and pass it to our algorithm.

3.0.0.1 Default Algorithms

We now provide two algorithms existing in R:

In clusterSVM and dcSVM, we offer an argument cluster.method, you could choose one of the two algorithms and pass its name to the argument.

3.0.0.2 Customization

We also offer arguments for users to pass their own implementation of the clustering algorithm: cluster.fun and cluster.predict.

cluster.fun is the clustering training function. It takes a function requiring the data and number of centers as the two main arguments. The output of this function should be an list object of the clustering result, while it has two fields:

  1. object$cluster as the clustering label on the input data.
  2. object$centers as the clustering center matrix.

cluster.predict is the predicting algorithm on the trained clustering object. It takes a function requiring the data and trained clustering object from cluster.fun as the two main arguments. The output of this function should be simply a vector of tue clustering label on the input data.

4 Clustered Support Vector Machines

4.0.0.1 Algorithm

The algorithm is straight forward:

Training

  1. Cluster the data. The default setting is stats::kmeans.
  2. Transform the data according to the Eq. (4) - Eq. (7) in the original paper.
  3. Solve the new problem with a linear svm from LiblineaR::LiblineaR.

Test

  1. Assign cluster label to each new data point, based on the clustering result from training.
  2. Transform the data according to the Eq. (4) - Eq. (7) in the original paper.
  3. Make prediction with the trained model.

4.0.0.2 Basic usage

We demonstrate the usage of this function with the following code:

## Time for Clustering: 0.00700000000000012 secs
## Time for Transforming: 0.0209999999999999 secs
## Time for Liblinear: 0.156 secs
## Time for Validation: 0.0110000000000001 secs
## 
## Total Time: 0.195 secs
## Accuracy Score: 0.81575
## [1] 0.81575

Here the parameters are grouped into four parts:

  1. x and y are the feature matric and target vector of the training data. type is specifying the mission and the type of the SVM.
  2. valid.x and valid.y are the feature matric and target vector of the validation data.
  3. seed is controlling the random seed to make the result reproducible. verbose is controlling the content of the output.
  4. centers is the parameter passing to the cluster algorithm.

Dense and sparse input

The sample data set is in the format of sparse matrix.

## [1] "matrix" "array"

The function takes a dense matrix or a sparse matrix as the input feature matrix. Therefore the following code gives you the same result.

## Time for Clustering: 0.00599999999999978 secs
## Time for Transforming: 0.00800000000000001 secs
## Time for Liblinear: 0.129 secs
## Time for Validation: 0.00800000000000001 secs
## 
## Total Time: 0.151 secs
## Accuracy Score: 0.81575
## [1] 0.81575

Self-defined clustering algorithm

In clusterSVM, the clustering is a very important step. Therefore we don’t restrict users to the RcppMLPACK::mlKmeans algorithm. Instead, we accept user-defined clustering algorithm as an argument.

Note that we require the output of the clustering algorithm contains two fields: centers and cluster. One example could be

Here we use the default kmeans, and implement the prediction function. Once we have defined the algorithm, it is straight forward to pass it to clusterSVM:

## Time for Clustering: 0.00600000000000023 secs
## Time for Transforming: 0.00300000000000011 secs
## Time for Liblinear: 0.131 secs
## Time for Validation: 0.00700000000000012 secs
## 
## Total Time: 0.148 secs
## Accuracy Score: 0.81575
## [1] 0.81575

5 A Divide-and-Conquer Support Vector Machine

5.0.0.1 Algorithm

The algorithm could be described as the following:

Training

  1. We cluster the data in a recursive manner:
    • Cluster the data in k groups.
    • In each groups, we keep cluster the data into k finer groups.
    • Repeat for max.levels times
  2. At the finest level, we train svm models
  3. At the j-th group on level l, we
    • Concatenate the alpha (coefficient on support vector) value from the subgroups of the j-th group
    • Train the svm model with the alpha value initialized
  4. Fine tune the alpha values by training an svm on all the support vectors of the whole data set.
  5. Train the final svm model with the alpha value

Test

There are two ways to do prediction.

  1. Early Prediction
    • In early predicton on level l, we predict the clustering label at level l for the new input data.
    • For data points belong to each subgroup, we make prediction using the corresponding svm model for this subgroup.
  2. Exact Prediction
    • We predict with the final svm model on the new input data.

5.0.0.2 Basic usage

We demonstrate the usage of this function with the following code:

## Finished clustering process in 0.0449999999999999 secs.
## Finished training in level 4
## Finished training in level 3
## Finished training in level 2
## Finished training in level 1
## Finished svm training process in 0.612 secs.
## Finished validation process in 0.0979999999999999 secs.
## Finished the whole process in 0.756 secs.
## [1] 0.81925

Here the parameters can be grouped into five parts.

  1. x and y are the feature matric and target vector of the training data.
  2. valid.x and valid.y are the feature matric and target vector of the validation data.
  3. seed is controlling the random seed to make the result reproducible.
  4. k, max.levels controls the size of the subproblem tree.
  5. early is the variable specifying whether we use early prediction or not. If early = 0 then we don’t use early prediction strategy, if early = l then we perform the early prediction at level l.

Early Prediction

We can do the early prediction by the following command:

## Finished clustering process in 0.00599999999999978 secs.
## Finished training in level 1
## Finished svm training process in 0.0720000000000001 secs.
## Finished validation process in 0.0179999999999998 secs.
## Finished the whole process in 0.0959999999999996 secs.
## [1] 0.86075
## elapsed 
##   0.096

It is faster because we can stop at a level, and don’t need to train SVMs for data of larger size.

Exact Prediction

To make the model more accurate, we can also perform the exact training by:

## Finished clustering process in 0.00599999999999978 secs.
## Finished training in level 1
## Finished svm training process in 0.0750000000000002 secs.
## Finished validation process in 0.0190000000000001 secs.
## Finished the whole process in 0.1 secs.
## [1] 0.85975
## elapsed 
##     0.1

This is more accurate but the time is longer. It a balance between the accuracy and the time complexity.

6 The Mixture of SVM Experts with a Gater Function

6.0.0.1 Algorithm

The algorithm can be described in the following iterative framework:

Training

  1. Split the data into \(M\) random subsets of nearly \(N/M\).
  2. Train \(M\) svm experts seperately on each subset.
  3. Each svm expert predict on the entire training data set.
  4. Train a neural network model minimizing the Eq. (7) in the original paper.
  5. Reconstruct subsets: Ffor each data point
    • sort the experts in descending order according to the weight from the neural network
    • assign the example to the first expert in the list which has less than (N/M+c) data points.
  6. Goto step 2 until a termination criterion is fulfilled.

Test

  1. Make prediction on the test data by the \(M\) experts.
  2. Make prediction on the experts’ output by the trained neural network model.

6.0.0.2 Basic usage

## Finished training for expert 1
## Finished training for expert 2
## Finished training for expert 3
## Finished training for expert 4
## Finished training for expert 5
## Finished training for expert 6
## Finished training for expert 7
## Finished training for expert 8
## Finished training for expert 9
## Finished training for expert 10
## Experts Training Time: 0.207
## 
Step: 1     Loss: 2.14497322        , threshold: 718.4687385    , min threshold: 288.9347304
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Step: 999   Loss: 0.2057398644      , threshold: 4.114715674    , min threshold: 1.069492498
## Finish gater training.
## Gater Neural Net Training Time: 3.454
## Finished training for expert 1
## Finished training for expert 2
## Finished training for expert 3
## Finished training for expert 4
## Finished training for expert 5
## Finished training for expert 6
## Finished training for expert 7
## Finished training for expert 8
## Finished training for expert 9
## Finished training for expert 10
## Experts Training Time: 0.173
## 
Step: 1     Loss: 0.5654535782      , threshold: 460.1478033    , min threshold: 15.14802317
Step: 2     Loss: 0.578643559       , threshold: 27.15821913    , min threshold: 15.14802317
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Step: 998   Loss: 0.1540599642      , threshold: 6.812435394    , min threshold: 2.902258797
Step: 999   Loss: 0.1540342125      , threshold: 7.922437305    , min threshold: 2.902258797
## Finish gater training.
## Gater Neural Net Training Time: 3.312
## Finished training for expert 1
## Finished training for expert 2
## Finished training for expert 3
## Finished training for expert 4
## Finished training for expert 5
## Finished training for expert 6
## Finished training for expert 7
## Finished training for expert 8
## Finished training for expert 9
## Finished training for expert 10
## Experts Training Time: 0.154
## 
Step: 1     Loss: 0.6097549704      , threshold: 122.0938077    , min threshold: 122.0938077
Step: 2     Loss: 0.5645530774      , threshold: 477.8983294    , min threshold: 122.0938077
Step: 3     Loss: 0.533065232       , threshold: 169.2139588    , min threshold: 122.0938077
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Step: 923   Loss: 0.1665409415      , threshold: 8.356566872    , min threshold: 2.574445474
Step: 924   Loss: 0.1664944292      , threshold: 9.470748116    , min threshold: 2.574445474
Step: 925   Loss: 0.1664417243      , threshold: 9.586427194    , min threshold: 2.574445474
Step: 926   Loss: 0.1663780315      , threshold: 9.989573836    , min threshold: 2.574445474
Step: 927   Loss: 0.1663058384      , threshold: 10.39813088    , min threshold: 2.574445474
Step: 928   Loss: 0.1662287983      , threshold: 9.244420477    , min threshold: 2.574445474
Step: 929   Loss: 0.1662112806      , threshold: 14.04768793    , min threshold: 2.574445474
Step: 930   Loss: 0.1662142084      , threshold: 29.7753883     , min threshold: 2.574445474
Step: 931   Loss: 0.1661301959      , threshold: 7.031050084    , min threshold: 2.574445474
Step: 932   Loss: 0.1660789783      , threshold: 11.28943132    , min threshold: 2.574445474
Step: 933   Loss: 0.166064568       , threshold: 21.72735002    , min threshold: 2.574445474
Step: 934   Loss: 0.1660100572      , threshold: 7.522536035    , min threshold: 2.574445474
Step: 935   Loss: 0.1659614803      , threshold: 5.930936455    , min threshold: 2.574445474
Step: 936   Loss: 0.165920083       , threshold: 12.62394391    , min threshold: 2.574445474
Step: 937   Loss: 0.1659033958      , threshold: 15.80649081    , min threshold: 2.574445474
Step: 938   Loss: 0.165860038       , threshold: 9.006152634    , min threshold: 2.574445474
Step: 939   Loss: 0.1658147833      , threshold: 8.11535029     , min threshold: 2.574445474
Step: 940   Loss: 0.1657628373      , threshold: 7.507249606    , min threshold: 2.574445474
Step: 941   Loss: 0.1657132683      , threshold: 7.338879216    , min threshold: 2.574445474
Step: 942   Loss: 0.165658153       , threshold: 7.253688285    , min threshold: 2.574445474
Step: 943   Loss: 0.1655954572      , threshold: 6.057081099    , min threshold: 2.574445474
Step: 944   Loss: 0.165528033       , threshold: 6.125220531    , min threshold: 2.574445474
Step: 945   Loss: 0.1654539301      , threshold: 6.819737012    , min threshold: 2.574445474
Step: 946   Loss: 0.1653753055      , threshold: 8.523131647    , min threshold: 2.574445474
Step: 947   Loss: 0.1653111273      , threshold: 10.90863419    , min threshold: 2.574445474
Step: 948   Loss: 0.1654490151      , threshold: 36.44959517    , min threshold: 2.574445474
Step: 949   Loss: 0.165245714       , threshold: 11.45400176    , min threshold: 2.574445474
Step: 950   Loss: 0.1652146822      , threshold: 12.34661112    , min threshold: 2.574445474
Step: 951   Loss: 0.1651614147      , threshold: 20.79586998    , min threshold: 2.574445474
Step: 952   Loss: 0.1651232187      , threshold: 12.25585764    , min threshold: 2.574445474
Step: 953   Loss: 0.1650837588      , threshold: 6.766919677    , min threshold: 2.574445474
Step: 954   Loss: 0.1650377269      , threshold: 8.395967174    , min threshold: 2.574445474
Step: 955   Loss: 0.1650191849      , threshold: 17.66023996    , min threshold: 2.574445474
Step: 956   Loss: 0.1649655928      , threshold: 7.358481074    , min threshold: 2.574445474
Step: 957   Loss: 0.1649165961      , threshold: 6.452833856    , min threshold: 2.574445474
Step: 958   Loss: 0.1648591921      , threshold: 6.429125092    , min threshold: 2.574445474
Step: 959   Loss: 0.1647959066      , threshold: 7.924913196    , min threshold: 2.574445474
Step: 960   Loss: 0.1647286536      , threshold: 10.08894632    , min threshold: 2.574445474
Step: 961   Loss: 0.1646743399      , threshold: 11.64584354    , min threshold: 2.574445474
Step: 962   Loss: 0.1646652567      , threshold: 20.12445089    , min threshold: 2.574445474
Step: 963   Loss: 0.1646138748      , threshold: 6.94923327     , min threshold: 2.574445474
Step: 964   Loss: 0.1645712086      , threshold: 7.990147838    , min threshold: 2.574445474
Step: 965   Loss: 0.164531397       , threshold: 12.94686358    , min threshold: 2.574445474
Step: 966   Loss: 0.1645092467      , threshold: 16.38856185    , min threshold: 2.574445474
Step: 967   Loss: 0.1644739139      , threshold: 9.662786545    , min threshold: 2.574445474
Step: 968   Loss: 0.1644328712      , threshold: 6.592660909    , min threshold: 2.574445474
Step: 969   Loss: 0.1643895905      , threshold: 5.138969242    , min threshold: 2.574445474
Step: 970   Loss: 0.1643435146      , threshold: 7.741587031    , min threshold: 2.574445474
Step: 971   Loss: 0.1642938727      , threshold: 6.638312513    , min threshold: 2.574445474
Step: 972   Loss: 0.1642484976      , threshold: 8.308507779    , min threshold: 2.574445474
Step: 973   Loss: 0.1641908757      , threshold: 5.305431342    , min threshold: 2.574445474
Step: 974   Loss: 0.1641285718      , threshold: 5.627610492    , min threshold: 2.574445474
Step: 975   Loss: 0.1640675838      , threshold: 8.703438529    , min threshold: 2.574445474
Step: 976   Loss: 0.1640508095      , threshold: 15.89875793    , min threshold: 2.574445474
Step: 977   Loss: 0.1639767975      , threshold: 8.420920118    , min threshold: 2.574445474
Step: 978   Loss: 0.1639188952      , threshold: 7.690176216    , min threshold: 2.574445474
Step: 979   Loss: 0.1639655678      , threshold: 10.77961109    , min threshold: 2.574445474
Step: 980   Loss: 0.1638628303      , threshold: 8.440022991    , min threshold: 2.574445474
Step: 981   Loss: 0.1638258159      , threshold: 5.173077937    , min threshold: 2.574445474
Step: 982   Loss: 0.163776472       , threshold: 4.381534515    , min threshold: 2.574445474
Step: 983   Loss: 0.1637549085      , threshold: 6.377977957    , min threshold: 2.574445474
Step: 984   Loss: 0.1637060998      , threshold: 9.703882338    , min threshold: 2.574445474
Step: 985   Loss: 0.1636932385      , threshold: 8.897098494    , min threshold: 2.574445474
Step: 986   Loss: 0.163655244       , threshold: 6.510910591    , min threshold: 2.574445474
Step: 987   Loss: 0.1636314234      , threshold: 8.8224256  , min threshold: 2.574445474
Step: 988   Loss: 0.1635909856      , threshold: 3.982896269    , min threshold: 2.574445474
Step: 989   Loss: 0.1635522457      , threshold: 6.884613703    , min threshold: 2.574445474
Step: 990   Loss: 0.1635177665      , threshold: 9.507298895    , min threshold: 2.574445474
Step: 991   Loss: 0.1634844487      , threshold: 7.182544856    , min threshold: 2.574445474
Step: 992   Loss: 0.1634562255      , threshold: 6.052425664    , min threshold: 2.574445474
Step: 993   Loss: 0.163436655       , threshold: 11.80264026    , min threshold: 2.574445474
Step: 994   Loss: 0.1634018673      , threshold: 6.660262075    , min threshold: 2.574445474
Step: 995   Loss: 0.1633618385      , threshold: 6.830521761    , min threshold: 2.574445474
Step: 996   Loss: 0.1633228174      , threshold: 6.048596278    , min threshold: 2.574445474
Step: 997   Loss: 0.1632760058      , threshold: 6.742892787    , min threshold: 2.574445474
Step: 998   Loss: 0.1632401424      , threshold: 7.972201953    , min threshold: 2.574445474
Step: 999   Loss: 0.1632021721      , threshold: 7.382592704    , min threshold: 2.574445474
## Finish gater training.
## Gater Neural Net Training Time: 3.43
## Finished validation process in 0.165000000000001 secs.
## Finished the whole process in 10.91 secs.
## [1] 0.9325

The parameters can be categorized into the following groups:

  1. x and y are the feature matric and target vector of the training data.
  2. valid.x and valid.y are the feature matric and target vector of the validation data.
  3. seed is controlling the random seed to make the result reproducible. verbose is controlling the content of the output.
  4. m, max.iter controls the iteration of “experts-gater” process.
  5. hidden, learningrate, threshold, stepmax are parameters for the neural network model.

7 Benchmarking

We offer benchmark codes to compare the performance and efficiency of our implementation. You can find the codes under inst/benchmark.

For some experiments running in a reasonable time, we have already generated some results. Those results are subjected to changes in the machine, system environment and the implementation.