Kernelized Budgeted Stochastic Gradient Descent

Support vector machines and other kernel-based learning algorithms are widely used and have many benefits. They can be considered as state-of-the-art algorithms in machine learning. Despite being easy to use, for larger data sets the kernelization, which was central to the development of SVM, becomes a bottleneck, as the computation time grows quadratically in the number of support vector– but the latter have been shown to grow linearly in the dataset size. Therefore the whole training process becomes quadratically, and impractical for even remotely large datasets. This problem was called the curse of kernelization in [WangCrammerVucetic2012].

There are different ways to solve this problem. One intuitive method was presented in [WangCrammerVucetic2012]. The idea is to put a constraint on the complexity of the model, i.e. the sparsity of the weight vector. As the weight vector in features space is a sum of basis functions, this means that it has to have the form \(w = \sum_{i=1}^B k(x_i, \cdot)\), where B is the chosen budget size and \(x_i\) are some data points.

[WangCrammerVucetic2012] employ a well-known stochastic gradient descent method, Pegasos, to train the model in a perceptron-like fashion: In each round the algorithms is given a data point. If it violates the margin with respect to the current model (so the example is either classified incorrectly or with a too low confidence), it will be added to the weight vector, also called budget, just like in Pegasos.

Obviously, at some point the budget becomes full. In this case, adding a new vector will violate the size-constraint. Therefore one needs a way to reduce the size of the budget. These, often heuristic, methods are called budget maintenence strategies. Many such strategies exist. One of the easiest is just to remove randomly a vector from the budget. Another strategy is remove the ‘oldest’ vector (this method is called Forgetron). Both strategies maintain the budget size, but are not optimal in a certain sense, as they do not really try to minimize the degradation of the model that occurs when one removes a support vector. A better way was proposed in [WangCrammerVucetic2012]: The idea is to find a pair of vectors that, when merged into one vector, does degrade the quality of the solution as low as possible. This can be formulated as an optimization problem, and it can be shown that with a heuristic search for such a pair, training is much better than with a random maintenence strategy.

In Shark both strategies, the remove and the merge strategy, can be applied. Tthis tutorial shows how to use the Kernelized Budgeted SGD Trainer in Shark with the merge strategy.

KernelBudgetedSGD in Shark

For this tutorial the following include files are needed:

#include <shark/Algorithms/Trainers/Budgeted/KernelBudgetedSGDTrainer.h> // the KernelBudgetedSGD trainer
#include <shark/Algorithms/Trainers/Budgeted/MergeBudgetMaintenanceStrategy.h> // the strategy the trainer will use
#include <shark/Data/DataDistribution.h> //includes small toy distributions
#include <shark/Models/Kernels/GaussianRbfKernel.h> //the used kernel for the SVM
#include <shark/ObjectiveFunctions/Loss/HingeLoss.h> // the loss we want to use for the SGD machine
#include <shark/ObjectiveFunctions/Loss/ZeroOneLoss.h> //used for evaluation of the classifier

Toy problem

In this tutorial, we consider the chessboard problem, which is a well-known artificial binary benchmark classification problem:

unsigned int ell = 500;     // number of training data point
unsigned int tests = 10000; // number of test data points

Chessboard problem; // artificial benchmark data
ClassificationDataset trainingData = problem.generateDataset(ell);
ClassificationDataset testData = problem.generateDataset(tests);

Model and learning algorithm

The steps to use the KernelBudgetedSGD trainer are the very same one uses to build a CSvmTrainer CSvmTrainer. Thus, to build our trainer, we need a KernelClassifier and an KernelBudgetedSGDTrainer.

Our model is given by the two components: A standard Gaussian/RBF kernel, which we create as usual:

double gamma = 0.5;         // kernel bandwidth parameter

GaussianRbfKernel<> kernel(gamma); // Gaussian kernel

and a kernel classifier:

KernelClassifier<RealVector> kernelClassifier; // (affine) linear function in kernel-induced feature space

Then, training the machine is simply performed by calling:

double C = 1.0;          // regularization parameter
bool bias = false;           // use bias/offset parameter
size_t budgetSize = 16;     // our model shall contain at most 16 vectors
size_t epochs = 5;      // we want to run 5 epochs

HingeLoss hingeLoss; // define the loss we want to use while training
// as the budget maintenance strategy we choose the merge strategy
MergeBudgetMaintenanceStrategy<RealVector> *strategy = new MergeBudgetMaintenanceStrategy<RealVector>();
KernelBudgetedSGDTrainer<RealVector> kernelBudgetedSGDtrainer(&kernel, &hingeLoss, C, bias, false, budgetSize, strategy);        // create the trainer
kernelBudgetedSGDtrainer.setEpochs(epochs);      // set the epochs number

As in the CSvmTrainer, the parameter C denotes the regularization parameter (the SVM uses the 1-norm penalty for target margin violations by default) and bias the inclusion of a bias term in the solver..

Evaluating the model

To evaluate the model, we simply create a test dataset by generating another chessboard problem. We can evaluate our trained model on the test data set as well as the train dataset (the latter one just to get a feeling how good the training went and to see overfitting problems). We consider the standard 0-1 loss as performance measure. The code then reads:

ZeroOneLoss<unsigned int> loss; // 0-1 loss
Data<unsigned int> output = kernelClassifier(trainingData.inputs()); // evaluate on training set
double train_error = loss.eval(trainingData.labels(), output);
cout << "training error:\t" <<  train_error << endl;
output = kernelClassifier(testData.inputs()); // evaluate on test set
double test_error = loss.eval(testData.labels(), output);
cout << "test error:\t" << test_error << endl;

Full example program

The full example program considered in this tutorial is KernelBudgetedSGDTutorial.cpp.


  1. Wang, K. Crammer and S. Vucetic: Breaking the curse of kernelization: Budgeted stochastic gradient descent for large-scale SVM training. The Journal of Machine Learning Research 13.1 (2012): 3103-3131.