Introduction

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

Gu, Q., & Han, J. (2013). Clustered support vector machines. In proceedings of the sixteenth international conference on artificial intelligence and statistics (pp. 307-315).
Hsieh, C. J., Si, S., & Dhillon, I. S. (2013). A divide-and-conquer solver for kernel support vector machines. arXiv preprint arXiv:1311.0914.
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

Reducing the scale of the data set results in a faster algorithm.
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.

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.

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.

Default Algorithms

We now provide two algorithms existing in R:

The default kmeans algorithm stats::kmeans, named as “kmeans”;
The kernel kmeans algorithm kernlab::kkmeans, named as “kernkmeans”.

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.

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:

object$cluster as the clustering label on the input data.
object$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.

Clustered Support Vector Machines

Algorithm

The algorithm is straight forward:

Training

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

Test

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

Basic usage

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

csvm.obj = clusterSVM(x = svmguide1[,-1], y = svmguide1[,1], type = 1,
                      valid.x = svmguide1.t[,-1],valid.y = svmguide1.t[,1], 
                      seed = 1, verbose = 1, centers = 8)

## Time for Clustering: 0.00700000000000012 secs

## Time for Transforming: 0.0219999999999998 secs

## Time for Liblinear: 0.184 secs

## Time for Validation: 0.0270000000000001 secs

## 
## Total Time: 0.242 secs

## Accuracy Score: 0.81575

csvm.obj$valid.score

## [1] 0.81575

Here the parameters are grouped into four parts:

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.
valid.x and valid.y are the feature matric and target vector of the validation data.
seed is controlling the random seed to make the result reproducible. verbose is controlling the content of the output.
centers is the parameter passing to the cluster algorithm.

Dense and sparse input

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

class(svmguide1)

## [1] "matrix"

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

csvm.obj = clusterSVM(x = as.matrix(svmguide1[,-1]), y = svmguide1[,1], type = 1,
                      valid.x = as.matrix(svmguide1.t[,-1]),valid.y = svmguide1.t[,1], 
                      seed = 1, verbose = 1, centers = 8)

## Time for Clustering: 0.00599999999999978 secs

## Time for Transforming: 0.004 secs

## Time for Liblinear: 0.2 secs

## Time for Validation: 0.0140000000000002 secs

## 
## Total Time: 0.225 secs

## Accuracy Score: 0.81575

csvm.obj$valid.score

## [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

cluster.fun = stats::kmeans

cluster.predict = function(x, cluster.object) {
  centers = cluster.object$centers
  eucliDist = function(x, centers) apply(centers, 1, function(C) colSums( (t(x)-C)^2 ))
  euclidean.dist = eucliDist(x, centers)
  result = max.col(-euclidean.dist)
  return(result)
}

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:

csvm.obj = clusterSVM(x = svmguide1[,-1], y = svmguide1[,1], centers = 8, seed = 1,
                      cluster.fun = cluster.fun, cluster.predict = cluster.predict,
                      valid.x = svmguide1.t[,-1], valid.y = svmguide1.t[,1])

## Time for Clustering: 0.0179999999999998 secs

## Time for Transforming: 0.04 secs

## Time for Liblinear: 0.325 secs

## Time for Validation: 0.0390000000000001 secs

## 
## Total Time: 0.425 secs

## Accuracy Score: 0.81575

csvm.obj$valid.score

## [1] 0.81575

A Divide-and-Conquer Support Vector Machine

Algorithm

The algorithm could be described as the following:

Training

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
At the finest level, we train svm models
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
Fine tune the alpha values by training an svm on all the support vectors of the whole data set.
Train the final svm model with the alpha value

Test

There are two ways to do prediction.

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.
Exact Prediction
- We predict with the final svm model on the new input data.

Basic usage

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

dcsvm.model = dcSVM(x = svmguide1[,-1], y = svmguide1[,1],
                    k = 4, max.levels = 4, seed = 0, cost = 32, gamma = 2,
                    kernel = 3,early = 0, m = 800,
                    valid.x = svmguide1.t[,-1], valid.y = svmguide1.t[,1])

## Finished clustering process in 0.0839999999999996 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 0.769 secs.

## Finished validation process in 0.0629999999999997 secs.

## Finished the whole process in 0.918 secs.

dcsvm.model$valid.score

## [1] 0.81925

Here the parameters can be grouped into five parts.

x and y are the feature matric and target vector of the training data.
valid.x and valid.y are the feature matric and target vector of the validation data.
seed is controlling the random seed to make the result reproducible.
k, max.levels controls the size of the subproblem tree.
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:

dcsvm.model = dcSVM(x = as.matrix(svmguide1[,-1]), y = svmguide1[,1], 
                    k = 10, max.levels = 1, 
                    early = 1, gamma = 2, cost = 32, tolerance = 1e-2, m = 800, 
                    valid.x = svmguide1.t[,-1], valid.y = svmguide1.t[,1])

## Finished clustering process in 0.00899999999999945 secs.

## Finished training in level 1

## Finished svm training process in 0.0990000000000002 secs.

## Finished validation process in 0.0219999999999994 secs.

## Finished the whole process in 0.130999999999999 secs.

dcsvm.model$valid.score

## [1] 0.86075

dcsvm.model$time$total.time

## elapsed 
##   0.131

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:

dcsvm.model = dcSVM(x = as.matrix(svmguide1[,-1]), y = svmguide1[,1], 
                    k = 10, max.levels = 1, 
                    early = 1, gamma = 2, cost = 32, tolerance = 1e-2, m = 800, 
                    valid.x = svmguide1.t[,-1], valid.y = svmguide1.t[,1])

## Finished clustering process in 0.00800000000000001 secs.

## Finished training in level 1

## Finished svm training process in 0.104 secs.

## Finished validation process in 0.0270000000000001 secs.

## Finished the whole process in 0.140000000000001 secs.

dcsvm.model$valid.score

## [1] 0.85975

dcsvm.model$time$total.time

## elapsed 
##    0.14

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

The Mixture of SVM Experts with a Gater Function

Algorithm

The algorithm can be described in the following iterative framework:

Training

Split the data into $M$ random subsets of nearly $N/M$.
Train $M$ svm experts seperately on each subset.
Each svm expert predict on the entire training data set.
Train a neural network model minimizing the Eq. (7) in the original paper.
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.
Goto step 2 until a termination criterion is fulfilled.

Test

Make prediction on the test data by the $M$ experts.
Make prediction on the experts' output by the trained neural network model.

Basic usage

gaterSVM.model = gaterSVM(x = svmguide1[,-1], y = svmguide1[,1], hidden = 10, seed = 0,
                          m = 10, max.iter = 3, learningrate = 0.01, threshold = 1, stepmax = 1000,
                          valid.x = svmguide1.t[,-1], valid.y = svmguide1.t[,1], verbose = TRUE)

## 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: 0.171

## 
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Step: 949   Loss: 0.2068861078      , threshold: 1.069492498    , min threshold: 1.069492498
Step: 950   Loss: 0.2068614532      , threshold: 2.278167011    , min threshold: 1.069492498
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Step: 998   Loss: 0.2057980289      , threshold: 38.89459237    , min threshold: 1.069492498
Step: 999   Loss: 0.2057398644      , threshold: 4.114715674    , min threshold: 1.069492498

## Finish gater training.

## Gater Neural Net Training Time: 6.046

## 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: 0.135

## 
Step: 1     Loss: 0.5654535782      , threshold: 460.1478033    , min threshold: 15.14802317
Step: 2     Loss: 0.578643559       , threshold: 27.15821913    , min threshold: 15.14802317
Step: 3     Loss: 0.5673487509      , threshold: 51.31314009    , min threshold: 15.14802317
Step: 4     Loss: 0.5111320699      , threshold: 59.30237623    , min threshold: 15.14802317
Step: 5     Loss: 0.4827015669      , threshold: 62.52028506    , min threshold: 15.14802317
Step: 6     Loss: 0.7228362388      , threshold: 1014.413063    , min threshold: 15.14802317
Step: 7     Loss: 0.4906929687      , threshold: 149.2156029    , min threshold: 15.14802317
Step: 8     Loss: 0.4781828663      , threshold: 272.593878     , min threshold: 15.14802317
Step: 9     Loss: 0.481466764       , threshold: 140.2767875    , min threshold: 15.14802317
Step: 10    Loss: 0.4625045607      , threshold: 44.24397049    , min threshold: 15.14802317
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Step: 986   Loss: 0.1543226149      , threshold: 11.65883182    , min threshold: 2.902258797
Step: 987   Loss: 0.1542681487      , threshold: 7.21906391     , min threshold: 2.902258797
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Step: 989   Loss: 0.1542131568      , threshold: 6.97533751     , min threshold: 2.902258797
Step: 990   Loss: 0.1541844785      , threshold: 13.453971  , min threshold: 2.902258797
Step: 991   Loss: 0.1542887431      , threshold: 38.04698349    , min threshold: 2.902258797
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Step: 993   Loss: 0.1541630144      , threshold: 25.12406019    , min threshold: 2.902258797
Step: 994   Loss: 0.1541382373      , threshold: 4.660828086    , min threshold: 2.902258797
Step: 995   Loss: 0.15411746        , threshold: 10.49815348    , min threshold: 2.902258797
Step: 996   Loss: 0.1541063961      , threshold: 3.233896044    , min threshold: 2.902258797
Step: 997   Loss: 0.1540838699      , threshold: 8.352716088    , min threshold: 2.902258797
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: 6.152

## 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: 0.111999999999998

## 
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
Step: 4     Loss: 0.5158132513      , threshold: 415.3989854    , min threshold: 122.0938077
Step: 5     Loss: 0.4834724336      , threshold: 185.9312012    , min threshold: 122.0938077
Step: 6     Loss: 0.4856721952      , threshold: 379.1479225    , min threshold: 122.0938077
Step: 7     Loss: 0.4633778571      , threshold: 45.96446228    , min threshold: 45.96446228
Step: 8     Loss: 0.4649908565      , threshold: 293.3698499    , min threshold: 45.96446228
Step: 9     Loss: 0.4536377533      , threshold: 23.2860236     , min threshold: 23.2860236
Step: 10    Loss: 0.4486952194      , threshold: 66.62881233    , min threshold: 23.2860236
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Step: 940   Loss: 0.1657628373      , threshold: 7.507249606    , min threshold: 2.574445474
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Step: 996   Loss: 0.1633228174      , threshold: 6.048596278    , min threshold: 2.574445474
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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: 6.785

## Finished validation process in 0.172000000000001 secs.

## Finished the whole process in 19.593 secs.

gaterSVM.model$valid.score

## [1] 0.9325

The parameters can be categorized into the following groups:

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

Benchmarking

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

utils.R contains helper functions to prepare the data.
preprocess_data.R contains codes that prepares the data for the other benchmark codes.
The other four files contains the benchmark codes:
- clustered_SVM.R contains simple codes for different data sets. clustered_SVM_Repeat.R contains repeated experiments and measures on the average performance of clusterSVM, and comparison against LiblineaR::LiblineaR and e1071::svm.
- dc_SVM.R contains experiments and measures on the performance of dcSVM, and comparison against e1071::svm.
- gater_SVM.R contains experiments and measures on the performance of gaterSVM, and comparison against e1071::svm.

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.