MapReduce is a programming model and an associated implementation for processing and generating large data sets. Users specify a map function that processes a key/value pair to generate a set of intermediate key/value pairs, and a reduce function that merges all intermediate values associated with the same intermediate key. Many real world tasks are expressible in this model, as shown in the paper.

Programs written in this functional style are automatically parallelized and executed on a large cluster of commodity machines. The run-time system takes care of the details of partitioning the input data, scheduling the program's execution across a set of machines, handling machine failures, and managing the required inter-machine communication. This allows programmers without any experience with parallel and distributed systems to easily utilize the resources of a large distributed system.

Our implementation of MapReduce runs on a large cluster of commodity machines and is highly scalable: a typical MapReduce computation processes many terabytes of data on thousands of machines. Programmers find the system easy to use: hundreds of MapReduce programs have been implemented and upwards of one thousand MapReduce jobs are executed on Google's clusters every day.

/pdf/mapreduce-simplified-data-processing-on-large-clusters-16emdgg5h5.pdf

MapReduce: simplified data processing on large clusters

MapReduce is a programming model and an associated implementation for processing and generating large datasets that is amenable to a broad variety of real-world tasks. Users specify the computation in terms of a map and a reduce function, and the underlying runtime system automatically parallelizes the computation across large-scale clusters of machines, handles machine failures, and schedules inter-machine communication to make efficient use of the network and disks. Programmers find the system easy to use: more than ten thousand distinct MapReduce programs have been implemented internally at Google over the past four years, and an average of one hundred thousand MapReduce jobs are executed on Google's clusters every day, processing a total of more than twenty petabytes of data per day.

/pdf/mapreduce-simplified-data-processing-on-large-clusters-1zth35tzla.pdf

Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.

/pdf/distributed-optimization-and-statistical-learning-via-the-8s9gtddw0d.pdf

Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers

We are given a large database of customer transactions. Each transaction consists of items purchased by a customer in a visit. We present an efficient algorithm that generates all significant association rules between items in the database. The algorithm incorporates buffer management and novel estimation and pruning techniques. We also present results of applying this algorithm to sales data obtained from a large retailing company, which shows the effectiveness of the algorithm.

/pdf/mining-association-rules-between-sets-of-items-in-large-13fvg5fahi.pdf

Mining association rules between sets of items in large databases

Publisher Summary This chapter provides an account of different neural network architectures for pattern recognition. A neural network consists of several simple processing elements called neurons. Each neuron is connected to some other neurons and possibly to the input nodes. Neural networks provide a simple computing paradigm to perform complex recognition tasks in real time. The chapter categorizes neural networks into three types: single-layer networks, multilayer feedforward networks, and feedback networks. It discusses the gradient descent and the relaxation method as the two underlying mathematical themes for deriving learning algorithms. A lot of research activity is centered on learning algorithms because of their fundamental importance in neural networks. The chapter discusses two important directions of research to improve learning algorithms: the dynamic node generation, which is used by the cascade correlation algorithm; and designing learning algorithms where the choice of parameters is not an issue. It closes with the discussion of performance and implementation issues.

Neural Networks for Pattern Recognition

A randomized algorithm that sorts on an N node network with constant valence in O(log N) time is given. More particularly, the algorithm sorts N items on an N-node cube-connected cycles graph, and, for some constant k, for all large enough a, it terminates within ka log N time with probability at least 1 - N-a.

/pdf/a-logarithmic-time-sort-for-linear-size-networks-dixccbqps4.pdf

A logarithmic time sort for linear size networks

Crytographic limitations on learning Boolean formulae and finite automata

Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in one-to-one, or possibly one-to-many, fashion. In this paper we propose a new kind of reduction that allows for gadgets with many-to-many correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a P-complete problem, then polynomial systems can be obtained, the solvability of which would imply P$^{{\tiny{\#P}}}$ = NC2.

Holographic Algorithms

https://people.seas.harvard.edu/%7Evaliant/bridging-2010.pdf

A bridging model for multi-core computing

We show that there is a combinational (acyclic) Boolean circuit of complexity 0(slog s), that can be made to compute any Boolean function of complexity s by setting its specially designated set of control inputs to appropriate fixed values. We investigate the construction of such “universal circuits” further so as to exhibit directions in which refinements of the asymptotic multiplicative constant factor in the complexity bound can be found. In this pursuit useful detailed guidance is provided by available lower bound arguments. In the final section we discuss some other problems in computational complexity that can be related directly to the graph-theoretic ideas behind our constructions. For motivation we start by illustrating some of the applications of universal circuits themselves.

Leslie G. Valiant

Papers

A logarithmic time sort for linear size networks

Crytographic limitations on learning Boolean formulae and finite automata

Holographic Algorithms

A bridging model for multi-core computing

Universal circuits (Preliminary Report)