This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.
Distributed optimization is a classical topic in applied mathematics with several excellent textbooks, e.g., @cite_15 @cite_1 @cite_34 .
@cite_15 @cite_34 @cite_1
2046387702 2542518093 2116137244 2556913586
We consider a network of agents that are cooperatively solving a global unconstrained optimization problem, where the objective function is the sum of privately known local objective functions of the agents. Recent literature on distributed optimization methods for solving this problem focused on subgradient based methods, which typically converge at the rate equation, where k is the number of iterations. In this paper, we introduce a new distributed optimization algorithm based on Alternating Direction Method of Multipliers (ADMM), which is a classical method for sequentially decomposing optimization problems with coupled constraints. We show that this algorithm converges at the rate equation. As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention in the literature. Although previous research has mostly focused on settings where either the observations, or features of the problem at hand are stored in distributed fashion, the situation where both are partitioned across the nodes of a computer cluster (doubly distributed) has barely been studied. In this work we propose two doubly distributed optimization algorithms. The first one falls under the umbrella of distributed dual coordinate ascent methods, while the second one belongs to the class of stochastic gradient coordinate descent hybrid methods. We conduct numerical experiments in Spark using real-world and simulated data sets and study the scaling properties of our methods. Our empirical evaluation of the proposed algorithms demonstrates the out-performance of a block distributed ADMM method, which, to the best of our knowledge is the only other existing doubly distributed optimization algorithm. We describe distributed algorithms for two widely-used topic models, namely the Latent Dirichlet Allocation (LDA) model, and the Hierarchical Dirichet Process (HDP) model. In our distributed algorithms the data is partitioned across separate processors and inference is done in a parallel, distributed fashion. We propose two distributed algorithms for LDA. The first algorithm is a straightforward mapping of LDA to a distributed processor setting. In this algorithm processors concurrently perform Gibbs sampling over local data followed by a global update of topic counts. The algorithm is simple to implement and can be viewed as an approximation to Gibbs-sampled LDA. The second version is a model that uses a hierarchical Bayesian extension of LDA to directly account for distributed data. This model has a theoretical guarantee of convergence but is more complex to implement than the first algorithm. Our distributed algorithm for HDP takes the straightforward mapping approach, and merges newly-created topics either by matching or by topic-id. Using five real-world text corpora we show that distributed learning works well in practice. For both LDA and HDP, we show that the converged test-data log probability for distributed learning is indistinguishable from that obtained with single-processor learning. Our extensive experimental results include learning topic models for two multi-million document collections using a 1024-processor parallel computer. The focus of this paper is the development and analysis of distributed algorithms for solving convex optimization problems that are defined over networks. Such network-structured optimization problems arise in a variety of application domains within the information sciences and engineering. For instance, problems such as multi-agent coordination, distributed tracking and localization, estimation problems in sensor networks and packet routing are all naturally cast as distributed convex minimization [2], [11], [10], [16], [20]. Common to these problems is the necessity for completely decentralized computation that is locally light — so as to avoid overburdening small sensors or flooding busy networks — and robust to periodic link or node failures. As a second example, data sets that are too large to be processed quickly by any single processor present related challenges. A canonical example that arises in statistical machine learning is the problem of minimizing a loss function averaged over a large dataset (e.g., optimization in support vector machines [5]). With terabytes of data, it is desirable to assign smaller subsets of the data to different processors, and the processors must communicate to find parameters that minimize the loss over the entire dataset. However, the communication should be efficient enough that network latencies do not offset computational gains.