The time complexity of support vector machines (SVMs) prohibits training on huge data sets with millions of samples. Recently, multilevel approaches to train SVMs have been developed to allow for time efficient training on huge data sets. While regular SVMs perform the entire training in one - time consuming - optimization step, multilevel SVMs first build a hierarchy of problems decreasing in size that resemble the original problem and then train an SVM model for each hierarchy level benefiting from the solved models of previous levels. We present a faster multilevel support vector machine that uses a label propagation algorithm to construct the problem hierarchy. Extensive experiments show that our new algorithm achieves speed-ups up to two orders of magnitude while having similar or better classification quality over state-of-the-art algorithms.
The support vector machine is a flexible optimization-based technique widely used for classification problems. In practice, its training part becomes computationally expensive on large-scale data sets because of such reasons as the complexity and number of iterations in parameter fitting methods, underlying optimization solvers, and nonlinearity of kernels. We introduce a fast multilevel framework for solving support vector machine models that is inspired by the algebraic multigrid. Significant improvement in the running has been achieved without any loss in the quality. The proposed technique is highly beneficial on imbalanced sets. We demonstrate computational results on publicly available and industrial data sets. The kernel support vector machine (SVM) is one of the most widely used classification methods; however, the amount of computation required becomes the bottleneck when facing millions of samples. In this paper, we propose and analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the division step, we partition the kernel SVM problem into smaller subproblems by clustering the data, so that each subproblem can be solved independently and efficiently. We show theoretically that the support vectors identified by the subproblem solution are likely to be support vectors of the entire kernel SVM problem, provided that the problem is partitioned appropriately by kernel clustering. In the conquer step, the local solutions from the subproblems are used to initialize a global coordinate descent solver, which converges quickly as suggested by our analysis. By extending this idea, we develop a multilevel Divide-and-Conquer SVM algorithm with adaptive clustering and early prediction strategy, which outperforms state-of-the-art methods in terms of training speed, testing accuracy, and memory usage. As an example, on the covtype dataset with half-a-million samples, DC-SVM is 7 times faster than LIBSVM in obtaining the exact SVM solution (to within @math relative error) which achieves 96.15 prediction accuracy. Moreover, with our proposed early prediction strategy, DC-SVM achieves about 96 accuracy in only 12 minutes, which is more than 100 times faster than LIBSVM. The support-vector network is a new learning machine for two-group classification problems. The machine conceptually implements the following idea: input vectors are non-linearly mapped to a very high-dimension feature space. In this feature space a linear decision surface is constructed. Special properties of the decision surface ensures high generalization ability of the learning machine. The idea behind the support-vector network was previously implemented for the restricted case where the training data can be separated without errors. We here extend this result to non-separable training data. High generalization ability of support-vector networks utilizing polynomial input transformations is demonstrated. We also compare the performance of the support-vector network to various classical learning algorithms that all took part in a benchmark study of Optical Character Recognition. The computational complexity of solving nonlinear support vector machine (SVM) is prohibitive on large-scale data. In particular, this issue becomes very sensitive when the data represents additional difficulties such as highly imbalanced class sizes. Typically, nonlinear kernels produce significantly higher classification quality to linear kernels but introduce extra kernel and model parameters which requires computationally expensive fitting. This increases the quality but also reduces the performance dramatically. We introduce a generalized fast multilevel framework for regular and weighted SVM and discuss several versions of its algorithmic components that lead to a good trade-off between quality and time. Our framework is implemented using PETSc which allows an easy integration with scientific computing tasks. The experimental results demonstrate significant speed up compared to the state-of-the-art nonlinear SVM libraries. Reproducibility: our source code, documentation and parameters are available at https: github.com esadr mlsvm.