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_6: 
  @cite_23: We present and study a distributed optimization algorithm by employing a stochastic dual coordinate ascent method. Stochastic dual coordinate ascent methods enjoy strong theoretical guarantees and often have better performances than stochastic gradient descent methods in optimizing regularized loss minimization problems. It still lacks of efforts in studying them in a distributed framework. We make a progress along the line by presenting a distributed stochastic dual coordinate ascent algorithm in a star network, with an analysis of the tradeoff between computation and communication. We verify our analysis by experiments on real data sets. Moreover, we compare the proposed algorithm with distributed stochastic gradient descent methods and distributed alternating direction methods of multipliers for optimizing SVMs in the same distributed framework, and observe competitive performances.
  @cite_17: Training machine learning models sometimes needs to be done on large amounts of data that exceed the capacity of a single machine, motivating recent works on developing algorithms that train in a distributed fashion. This paper proposes an efficient box-constrained quadratic optimization algorithm for distributedly training linear support vector machines (SVMs) with large data. Our key technical contribution is an analytical solution to the problem of computing the optimal step size at each iteration, using an efficient method that requires only O(1) communication cost to ensure fast convergence. With this optimal step size, our approach is superior to other methods by possessing global linear convergence, or, equivalently, O(log(1 e)) iteration complexity for an e-accurate solution, for distributedly solving the non-strongly-convex linear SVM dual problem. Experiments also show that our method is significantly faster than state-of-the-art distributed linear SVM algorithms including DSVM-AVE, DisDCA and TRON.
summary: On the theoretical front: previous papers @cite_23 @cite_6 can only show linear convergence when the objective function is @math and @math is strongly convex. @cite_17 considered some non-strongly convex functions, but they assume each subproblem is solved exactly. In this paper, we prove a global linear convergence even when @math is not strongly convex and each subproblem is solved approximately. Our proof covers general DBCD algorithms for some non-strongly convex functions, for which previous analysis can only show sub-linear convergence.