text: We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.
  @cite_23: For supervised classification problems, it is well known that learnability is equivalent to uniform convergence of the empirical risks and thus to learnability by empirical minimization. Inspired by recent regret bounds for online convex optimization, we study stochastic convex optimization, and uncover a surprisingly different situation in the more general setting: although the stochastic convex optimization problem is learnable (e.g. using online-to-batch conversions), no uniform convergence holds in the general case, and empirical minimization might fail. Rather then being a difference between online methods and a global minimization approach, we show that the key ingredient is strong convexity and regularization. Our results demonstrate that the celebrated theorem of on the equivalence of learnability and uniform convergence does not extend to Vapnik’s General Setting of Learning, that in the General Setting considering only empirical minimization is not enough, and that despite Vanpnik’s result on the equivalence of strict consistency and uniform convergence, uniform convergence is only a sufficient, but not necessary, condition for meaningful non-trivial learnability.
  @cite_3: We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably improves with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.
  @cite_10: 
  @cite_6: We consider distributed convex optimization problems originating from sample average approximation of stochastic optimization, or empirical risk minimization in machine learning. We assume that each machine in the distributed computing system has access to a local empirical loss function, constructed with i.i.d. data sampled from a common distribution. We propose a communication-efficient distributed algorithm to minimize the overall empirical loss, which is the average of the local empirical losses. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the results for ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning where the condition number of the problem grows with square root of the sample size, the required number of communication rounds of the algorithm does not increase with the sample size, and only grows slowly with the number of machines.
summary: In Table , we compare the rounds and runtime of DSVRG, DASVRG and DISCO in the case where @math , which is a typical setting for ERM problem as justified in @cite_6 @cite_3 @cite_10 @cite_23 . We only compare our methods against DISCO, since it uses the fewest rounds of communication among other related algorithms. Let us consider the case where @math , which is true in almost any reasonable distributed computing scenario. We can see from Table that the rounds needed by DSVRG is lower than that of DISCO. In fact, both DSVRG and DASVRG use @math rounds of communication, which is almost a constant for many practical machine learning applications Typically @math , so @math is always less than @math . . .