This paper concerns the distributed training of nonlinear kernel machines on Map-Reduce. We show that a re-formulation of Nystr "om approximation based solution which is solved using gradient based techniques is well suited for this, especially when it is necessary to work with a large number of basis points. The main advantages of this approach are: avoidance of computing the pseudo-inverse of the kernel sub-matrix corresponding to the basis points; simplicity and efficiency of the distributed part of the computations; and, friendliness to stage-wise addition of basis points. We implement the method using an AllReduce tree on Hadoop and demonstrate its value on a few large benchmark datasets.
There are methods @cite_11 @cite_18 that partition the full data into several nodes, obtain a solution on each partition and combine them centrally. This ensemble solution is usually very sub-optimal. Some methods @cite_3 @cite_30 @cite_8 @cite_29 @cite_13 do further processing of those solutions to come closer to full optimality, but this requires heavy communication of data, e.g., passing support vector data between nodes. There are methods that directly parallelize well-known sequential methods such as SMO and SVMlight @cite_0 @cite_16 , but these involve communication costs over a large number of iterations.
@cite_30 @cite_18 @cite_8 @cite_29 @cite_3 @cite_0 @cite_16 @cite_13 @cite_11
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We consider the partitioning of a problem on a domain with unequal work estimates in different subdomains in a way that balances the workload across multiple processors. Such a problem arises for example in solving partial differential equations using an adaptive method that places extra grid points in certain subregions of the domain. We use a binary decomposition of the domain to partition it into rectangles requiring equal computational effort. We then study the communication costs of mapping this partitioning onto different multiprocessors: a mesh- connected array, a tree machine, and a hypercube. The communication cost expressions can be used to determine the optimal depth of the above partitioning. Abstract Hierarchical N-body methods, which are based on a fundamental insight into the nature of many physical processes, are increasingly being used to solve large-scale problems in a variety of scientific engineering domains. Applications that use these methods are challenging to parallelize effectively, however, owing to their nonuniform, dynamically changing characteristics and their need for long-range communication. In this paper, we study the partitioning and scheduling techniques required to obtain effective parallel performance on applications that use a range of hierarchical N-body methods. To obtain representative coverage, we first examine applications that use the two best methods known for classical N-body problems: the Barnes-Hut method and the fast multipole method. Then, we examine a recent hierarchical method for radiosity calculations in computer graphics, which applies the hierarchical N-body approach to a problem with very different characteristics. We find that straightforward decomposition techniques which an automatic scheduler might implement do not scale well, because they are unable to simultaneously provide load balancing and data locality. However, all the applications yield very good parallel performance if appropriate partitioning and scheduling techniques are implemented by the programmer. For the applications that use the Barnes-Hut and fast multipole methods, simple yet effective partitioning techniques can be developed by exploiting some key insights into both the methods and the classical problems that they solve. Using a novel partitioning technique, even relatively small problems achieve 45-fold speedups on a 48-processor Stanford DASH machine (a cache-coherent, shared address space multiprocessor) and 118-fold speedups on a 128-processor simulated architecture. The very different characteristics of the radiosity application require a different partitioning scheduling approach to be used for it; however, it too yields very good parallel performance. In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms) is that it partitions the vertices of the graph intopparts while distributing the overall adjacency matrix of the graph among allpprocessors. This mapping results in substantially smaller communication than one-dimensional distribution for graphs with relatively high degree, especially if the graph is randomly distributed among the processors. We also present a parallel algorithm for computing a minimal cover of a bipartite graph which is a key operation for obtaining a small vertex separator that is useful for computing the fill reducing ordering of sparse matrices. Our parallel algorithm achieves a speedup of up to 56 on 128 processors for moderate size problems, further reducing the already moderate serial run time of multilevel schemes. Furthermore, the quality of the produced partitions and orderings are comparable to those produced by the serial multilevel algorithm that has been shown to outperform both spectral partitioning and multiple minimum degree. Distributed graph processing platforms have helped emerging application domains use commodity clusters and Clouds to analyze large graphs. Vertex-centric programming models like Google Pregel, and their subgraph-centric variants, specify data-parallel application logic for a single vertex or component that execute iteratively. The locality and balancing of components within partitions affects the performance of such platforms. We propose three partitioning strategies for a subgraph-centric model, and analyze their impact on CPU utilization, communication, iterations, and makespan. We analyze these using Breadth First Search and PageRank algorithms on powerlaw and spatio-planar graphs. They are validated on a commodity cluster using our GoFFish subgraph-centric platform, and compared against Apache Giraph vertex-centric platform. Our experiments show upto 8 times improvement in utilization resulting to upto 5 times improvement of overall makespan for flat and hierarchical partitioning over the default strategy due to improved machine utilization. Further, these also exhibit better horizontal scalability relative to Giraph. Parallel database systems horizontally partition large amounts of structured data in order to provide parallel data processing capabilities for analytical workloads in shared-nothing clusters. One major challenge when horizontally partitioning large amounts of data is to reduce the network costs for a given workload and a database schema. A common technique to reduce the network costs in parallel database systems is to co-partition tables on their join key in order to avoid expensive remote join operations. However, existing partitioning schemes are limited in that respect since only subsets of tables in complex schemata sharing the same join key can be co-partitioned unless tables are fully replicated. In this paper we present a novel partitioning scheme called predicate-based reference partition (or PREF for short) that allows to co-partition sets of tables based on given join predicates. Moreover, based on PREF, we present two automatic partitioning design algorithms to maximize data-locality. One algorithm only needs the schema and data whereas the other algorithm additionally takes the workload as input. In our experiments we show that our automated design algorithms can partition database schemata of different complexity and thus help to effectively reduce the runtime of queries under a given workload when compared to existing partitioning approaches. We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss. Consider a complete communication network on n nodes. In synchronous 2-counting, the nodes receive a common clock pulse and they have to agree on which pulses are "odd" and which are "even". Furthermore, the solution needs to be self-stabilising (reaching correct operation from any initial state) and tolerate f Byzantine failures (nodes that send arbitrary misinformation). Prior algorithms either require a source of random bits or a large number of states per node. In this work, we give fast state-optimal deterministic algorithms for the first non-trivial case f = 1 . To obtain these algorithms, we develop and evaluate two different techniques for algorithm synthesis. Both are based on casting the synthesis problem as a propositional satisfiability (SAT) problem; a direct encoding is efficient for synthesising time-optimal algorithms, while an approach based on counter-example guided abstraction refinement discovers non-optimal algorithms quickly. We develop computational techniques to find algorithms for synchronous 2-counting.Automated synthesis yields state-optimal self-stabilising fault-tolerant algorithms.We give a thorough experimental comparison of our two SAT-based synthesis techniques.A direct SAT encoding is more efficient for finding time-optimal algorithms.An iterative CEGAR-based approach finds non-optimal algorithms quickly. In recent years, Massively Parallel Processors (MPPs) have gained ground enabling vast amounts of data processing. In such environments, data is partitioned across multiple compute nodes, which results in dramatic performance improvements during parallel query execution. To evaluate certain relational operators in a query correctly, data sometimes needs to be re-partitioned (i.e., moved) across compute nodes. Since data movement operations are much more expensive than relational operations, it is crucial to design a suitable data partitioning strategy that minimizes the cost of such expensive data transfers. A good partitioning strategy strongly depends on how the parallel system would be used. In this paper we present a partitioning advisor that recommends the best partitioning design for an expected workload. Our tool recommends which tables should be replicated (i.e., copied into every compute node) and which ones should be distributed according to specific column(s) so that the cost of evaluating similar workloads is minimized. In contrast to previous work, our techniques are deeply integrated with the underlying parallel query optimizer, which results in more accurate recommendations in a shorter amount of time. Our experimental evaluation using a real MPP system, Microsoft SQL Server 2008 Parallel Data Warehouse, with both real and synthetic workloads shows the effectiveness of the proposed techniques and the importance of deep integration of the partitioning advisor with the underlying query optimizer. In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed to approximately solve the problem with high probability, is a simple expression depending on the number of parallel processors and a natural and easily computable measure of separability of the smooth component of the objective function. In the worst case, when no degree of separability is present, there may be no speedup; in the best case, when the problem is separable, the speedup is equal to the number of processors. Our analysis also works in the mode when the number of blocks being updated at each iteration is random, which allows for modeling situations with busy or unreliable processors. We show that our algorithm is able to solve a LASSO problem involving a matrix with 20 billion nonzeros in 2 h on a large memory node with 24 cores.