This work proposes a progressive manifold identification approach for distributed optimization with sound theoretical justifications to greatly reduce both the rounds of communication and the bytes communicated per round for partly-smooth regularized problems such as the $\ell_1$- and group-LASSO-regularized ones. Our two-stage method first uses an inexact proximal quasi-Newton method to iteratively identify a sequence of low-dimensional manifolds in which the final solution would lie, and restricts the model update within the current manifold to gradually lower the order of the per-round communication cost from the problem dimension to the dimension of the manifold that contains a solution and makes the problem within it smooth. After identifying this manifold, we take superlinear-convergent truncated semismooth Newton steps computed by preconditioned conjugate gradient to largely reduce the communication rounds by improving the convergence rate from the existing linear or sublinear ones to a superlinear rate. Experiments show that our method can be orders of magnitudes lower in the communication cost and an order of magnitude faster in the running time than the state of the art.

Abstract In recent years, there is a growing need to train machine learning models on a huge volume of data. Therefore, designing efficient distributed optimization algorithms for empirical risk minimization (ERM) has become an active and challenging research topic. In this paper, we propose a flexible framework for distributed ERM training through solving the dual problem, which provides a unified description and comparison of existing methods. Our approach requires only approximate solutions of the sub-problems involved in the optimization process, and is versatile to be applied on many large-scale machine learning problems including classification, regression, and structured prediction. We show that our framework enjoys global linear convergence for a broad class of non-strongly-convex problems, and some specific choices of the sub-problems can even achieve much faster convergence than existing approaches by a refined analysis. This improved convergence rate is also reflected in the superior empirical performance of our method.

State-of-the-art first- and second-order optimization methods are able to achieve either fast global linear convergence rates or quadratic convergence, but not both of them. In this work, we propose an interpolation between first- and second-order methods for regularized empirical risk minimization that exploits the problem structure to efficiently combine multiple update directions. Our method attains both optimal global linear convergence rate for first-order methods, and local quadratic convergence. Experimental results show that our method outperforms state-of-the-art first- and second-order optimization methods in terms of the number of data accesses, while is competitive in training time.

For distributed linear classification, L1 regularization is useful because of a smaller model size. However, with the non-differentiability, it is more difficult to develop efficient optimization algorithms. In the past decade, OWLQN has emerged as the major method for distributed training of L1 problems. In this work, we point out issues in OWLQN's search directions. Then we extend the recently developed limited-memory common-directions method for L2-regularized problems to L1 scenarios. Through a unified interpretation of batch methods for L1 problems, we explain why OWLQN has been a popular method and why our method is superior in distributed environments. Experiments confirm that the proposed method is faster than OWLQN in most situations.

Distributed optimization has become an important research topic for dealing with extremely large volume of data available in the Internet companies nowadays. Additional machines make computation less expensive, but inter-machine communication becomes prominent in the optimization process, and efficient optimization methods should reduce the amount of the communication in order to achieve shorter overall running time. In this work, we utilize the advantages of the recently proposed, theoretically fast-convergent common-directions method, but tackle its main drawback of excessive spatial and computational costs to propose a limited-memory algorithm. The result is an efficient, linear-convergent optimization method for paraliel/distributed optimization. We further discuss how our method can exploit the problem structure to efficiently train regularized empirical risk minimization (ERM) models. Experimental results show that our method outperforms state-of-the-art distributed optimization methods for ERM problems.

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/ε)) iteration complexity for an epsilon-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.

Logistic regression and linear SVM are useful methods for large-scale classification. However, their distributed implementations have not been well studied. Recently, because of the inefficiency of the MapReduce framework on iterative algorithms, Spark, an in-memory cluster-computing platform, has been proposed. It has emerged as a popular framework for large-scale data processing and analytics. In this work, we consider a distributed Newton method for solving logistic regression as well linear SVM and implement it on Spark. We carefully examine many implementation issues significantly affecting the running time and propose our solutions. After conducting thorough empirical investigations, we release an efficient and easy-to-use tool for the Spark community.

Learning to rank is an important task for recommendation systems, online advertisement and web search. Among those learning to rank methods, rankSVM is a widely used model. Both linear and nonlinear (kernel) rankSVM have been extensively studied, but the lengthy training time of kernel rankSVM remains a challenging issue. In this paper, after discussing difficulties of training kernel rankSVM, we propose an efficient method to handle these problems. The idea is to reduce the number of variables from quadratic to linear with respect to the number of training instances, and efficiently evaluate the pairwise losses. Our setting is applicable to a variety of loss functions. Further, general optimization methods can be easily applied to solve the reformulated problem. Implementation issues are also carefully considered. Experiments show that our method is faster than state-of-the-art methods for training kernel rankSVM.

Linear rankSVM is one of the widely used methods for learning to rank. Although its performance may be inferior to nonlinear methods such as kernel rankSVM and gradient boosting decision trees, linear rankSVM is useful to quickly produce a baseline model. Furthermore, following its recent development for classification, linear rankSVM may give competitive performance for large and sparse data. A great deal of works have studied linear rankSVM. The focus is on the computational efficiency when the number of preference pairs is large. In this letter, we systematically study existing works, discuss their advantages and disadvantages, and propose an efficient algorithm. We discuss different implementation issues and extensions with detailed experiments. Finally, we develop a robust linear rankSVM tool for public use.

Crammer and Singer's method is one of the most popular multiclass support vector machines (SVMs). It considers L1 loss (hinge loss) in a complicated optimization problem. In SVM, squared hinge loss (L2 loss) is a common alternative to L1 loss, but surprisingly we have not seen any paper studying the details of Crammer and Singer's method using L2 loss. In this letter, we conduct a thorough investigation. We show that the derivation is not trivial and has some subtle differences from the L1 case. Details provided in this work can be a useful reference for those who intend to use Crammer and Singer's method with L2 loss. They do not need a tedious process to derive everything by themselves. Furthermore, we present some new results on and discussion of both L1- and L2-loss formulations.