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.
Building on the work of @cite_19 , we prove a new lower bound showing that any distributed first-order algorithm requires @math rounds. This lower bound combined with the convergence analysis of DASVRG shows that DASVRG is optimal in the number of rounds of communication.
@cite_19
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We show that any randomised Monte Carlo distributed algorithm for the Lov 'asz local lemma requires @math communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of @math , where @math is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lov 'asz local lemma with a running time of @math rounds in bounded-degree graphs, and the best lower bound before our work was @math rounds [ 2014]. We present a near-optimal distributed algorithm for @math -approximation of single-commodity maximum flow in undirected weighted networks that runs in @math communication rounds in the model. Here, @math and @math denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of @math , and it nearly matches the @math round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A @math -round distributed construction of a spanning tree of average stretch @math . (ii) A @math -round distributed construction of an @math -congestion approximator consisting of the cuts induced by @math virtual trees. The distributed representation of the cut approximator allows for evaluation in @math rounds. All our algorithms make use of randomization and succeed with high probability. In this paper, we prove a lower bound on the number of rounds required by a deterministic distributed protocol for broadcasting a message in radio networks whose processors do not know the identities of their neighbors. Such an assumption captures the main characteristic of mobile and wireless environments [3], i.e., the instability of the network topology. For any distributed broadcast protocol II, for any n and for any D ≤ n 2, we exhibit a network G with n nodes and diameter D such that the number of rounds needed by H for broadcasting a message in G is Ω(D log n). The result still holds even if the processors in the network use a different program and know n and D. We also consider the version of the broadcast problem in which an arbitrary number of processors issue at the same time an identical message that has to be delivered to the other processors. In such a case we prove that, even assuming that the processors know the network topology, Ω(n) rounds are required for solving the problem on a complete network (D=1) with n processors. We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.