text: Kernel machines often yield superior predictive performance on various tasks; however, they suffer from severe computational challenges. In this paper, we show how to overcome the important challenge of speeding up kernel machines. In particular, we develop a parallel block minimization framework for solving kernel machines, including kernel SVM and kernel logistic regression. Our framework proceeds by dividing the problem into smaller subproblems by forming a block-diagonal approximation of the Hessian matrix. The subproblems are then solved approximately in parallel. After that, a communication efficient line search procedure is developed to ensure sufficient reduction of the objective function value at each iteration. We prove global linear convergence rate of the proposed method with a wide class of subproblem solvers, and our analysis covers strongly convex and some non-strongly convex functions. We apply our algorithm to solve large-scale kernel SVM problems on distributed systems, and show a significant improvement over existing parallel solvers. As an example, on the covtype dataset with half-a-million samples, our algorithm can obtain an approximate solution with 96 accuracy in 20 seconds using 32 machines, while all the other parallel kernel SVM solvers require more than 2000 seconds to achieve a solution with 95 accuracy. Moreover, our algorithm can scale to very large data sets, such as the kdd algebra dataset with 8 million samples and 20 million features.
  @cite_27: A major problem for kernel-based predictors (such as Support Vector Machines and Gaussian processes) is that the amount of computation required to find the solution scales as O(n3), where n is the number of training examples. We show that an approximation to the eigendecomposition of the Gram matrix can be computed by the Nystrom method (which is used for the numerical solution of eigenproblems). This is achieved by carrying out an eigendecomposition on a smaller system of size m < n, and then expanding the results back up to n dimensions. The computational complexity of a predictor using this approximation is O(m2n). We report experiments on the USPS and abalone data sets and show that we can set m ≪ n without any significant decrease in the accuracy of the solution.
  @cite_16: To accelerate the training of kernel machines, we propose to map the input data to a randomized low-dimensional feature space and then apply existing fast linear methods. The features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shift-invariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in large-scale classification and regression tasks linear machine learning algorithms applied to these features outperform state-of-the-art large-scale kernel machines.
  @cite_13: We demonstrate that distributed block coordinate descent can quickly solve kernel regression and classification problems with millions of data points. Armed with this capability, we conduct a thorough comparison between the full kernel, the Nystr "om method, and random features on three large classification tasks from various domains. Our results suggest that the Nystr "om method generally achieves better statistical accuracy than random features, but can require significantly more iterations of optimization. Lastly, we derive new rates for block coordinate descent which support our experimental findings when specialized to kernel methods.
  @cite_20: Despite their theoretical appeal and grounding in tractable convex optimization techniques, kernel methods are often not the first choice for large-scale speech applications due to their significant memory requirements and computational expense. In recent years, randomized approximate feature maps have emerged as an elegant mechanism to scale-up kernel methods. Still, in practice, a large number of random features is required to obtain acceptable accuracy in predictive tasks. In this paper, we develop two algorithmic schemes to address this computational bottleneck in the context of kernel ridge regression. The first scheme is a specialized distributed block coordinate descent procedure that avoids the explicit materialization of the feature space data matrix, while the second scheme gains efficiency by combining multiple weak random feature models in an ensemble learning framework. We demonstrate that these schemes enable kernel methods to match the performance of state of the art Deep Neural Networks on TIMIT for speech recognition and classification tasks. In particular, we obtain the best classification error rates reported on TIMIT using kernel methods.
summary: Kernel approximation approaches. Another line of approach aims to approximate the kernel matrix and then solve the reduced size problem. Nystr " o m approximation @cite_27 and random Fourier features @cite_16 can be used to form low-rank approximation of kernel matrices, and then the problem can be reduced to a linear SVM or logistic regression problem. To parallelize these algorithms, a distributed linear SVM or logistic regression solver is needed. The comparisons are in Figure . Using random Fourier features in kernel machine is also applied in @cite_20 and @cite_13 . Comparing with our proposed method, (1) they only consider solving linear systems, and it is nontrivial for them to solve kernel SVM and logistic regression problems; (2) they cannot obtain the exact solution of kernel machines, since they only use the approximate kernel.