Approximate Stochastic Subgradient Estimation Training for Support Vector Machines

Subgradient algorithms for training support vector machines have been quite successful for solving large-scale and online learning problems. However, they have been restricted to linear kernels and strongly convex formulations. This paper describes efficient subgradient approaches without such limitations. Our approaches make use of randomized low-dimensional approximations to nonlinear kernels, and minimization of a reduced primal formulation using an algorithm based on robust stochastic approximation, which do not require strong convexity. Experiments illustrate that our approaches produce solutions of comparable prediction accuracy with the solutions acquired from existing SVM solvers, but often in much shorter time. We also suggest efficient prediction schemes that depend only on the dimension of kernel approximation, not on the number of support vectors.Comment: An extended version of the ICPRAM 2012 pape

Distributed Inference for Linear Support Vector Machine

The growing size of modern data brings many new challenges to existing statistical inference methodologies and theories, and calls for the development of distributed inferential approaches. This paper studies distributed inference for linear support vector machine (SVM) for the binary classification task. Despite a vast literature on SVM, much less is known about the inferential properties of SVM, especially in a distributed setting. In this paper, we propose a multi-round distributed linear-type (MDL) estimator for conducting inference for linear SVM. The proposed estimator is computationally efficient. In particular, it only requires an initial SVM estimator and then successively refines the estimator by solving simple weighted least squares problem. Theoretically, we establish the Bahadur representation of the estimator. Based on the representation, the asymptotic normality is further derived, which shows that the MDL estimator achieves the optimal statistical efficiency, i.e., the same efficiency as the classical linear SVM applying to the entire data set in a single machine setup. Moreover, our asymptotic result avoids the condition on the number of machines or data batches, which is commonly assumed in distributed estimation literature, and allows the case of diverging dimension. We provide simulation studies to demonstrate the performance of the proposed MDL estimator.Comment: 50 pages, 11 figure