In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. Here, we review recent work on developing and implementing randomized matrix algorithms in large-scale parallel and distributed environments. Randomized algorithms for matrix problems have received a great deal of attention in recent years, thus far typically either in theory or in machine learning applications or with implementations on a single machine. Our main focus is on the underlying theory and practical implementation of random projection and random sampling algorithms for very large very overdetermined (i.e., overconstrained) $\ell_1$ and $\ell_2$ regression problems. Randomization can be used in one of two related ways: either to construct sub-sampled problems that can be solved, exactly or approximately, with traditional numerical methods; or to construct preconditioned versions of the original full problem that are easier to solve with traditional iterative algorithms. Theoretical results demonstrate that in near input-sparsity time and with only a few passes through the data one can obtain very strong relative-error approximate solutions, with high probability. Empirical results highlight the importance of various trade-offs (e.g., between the time to construct an embedding and the conditioning quality of the embedding, between the relative importance of computation versus communication, etc.) and demonstrate that $\ell_1$ and $\ell_2$ regression problems can be solved to low, medium, or high precision in existing distributed systems on up to terabyte-sized data.
We present a new programming paradigm which can be useful, in particular, for implementing window interfaces and parallel algorithms. This paradigm allows a user to define operators which can contain nested operators. The new paradigm is called operator-oriented. One of the goals of this paradigm is to escape the complexity of objects definitions inherent in many object-oriented languages and to move to transparent algorithms definitions.
We analyze the parallel performance of randomized interpolative decomposition by decomposing low rank complex-valued Gaussian random matrices up to 64 GB. We chose a Cray XMT supercomputer as it provides an almost ideal PRAM model permitting quick investigation of parallel algorithms without obfuscation from hardware idiosyncrasies. We obtain that on non-square matrices performance becomes very good, with overall runtime over 70 times faster on 128 processors. We also verify that numerically discovered error bounds still hold on matrices nearly two orders of magnitude larger than those previou…
This work explores the characteristics of implementing parallel Quick Sort algorithm over the OTIS Hyper Hexa-Cell interconnection network OHHC. OHHC interconnection architecture offers efficient processor connectivity by utilizing both electronic and optical based connections. The work presented includes analytical evaluation of the algorithm as well as simulated evaluation over a multi-threading environment. Different experiments were performed using different OHHC dimensions, different integer array types and different array sizes. The evaluation and simulation demonstrated encouraging resu…
University of California, Berkeley（加州大学伯克利分校）；ETH Zürich（苏黎世联邦理工学院）
We study preconditioned gradient-based optimization methods where the preconditioning matrix has block-diagonal form. Such a structural constraint comes with the advantage that the update computation is block-separable and can be parallelized across multiple independent tasks. Our main contribution is to demonstrate that the convergence of these methods can significantly be improved by a randomization technique which corresponds to repartitioning coordinates across tasks during the optimization procedure. We provide a theoretical analysis that accurately characterizes the expected convergence …
We present a set of programming tools (classes and functions written in C++ and based on Message Passing Interface) for fast development of generic parallel (and non-parallel) lattice simulations. They are collectively called MDP 1.2. These programming tools include classes and algorithms for matrices, random number generators, distributed lattices (with arbitrary topology), fields and parallel iterations. No previous knowledge of MPI is required in order to use them. Some applications in electromagnetism, electronics, condensed matter and lattice QCD are presented.
Motivated by recent distributed systems technology, Aguilera et al. introduced a hybrid model of distributed computing, called message-and-memory model or m&m model for short [1]. In this model, processes can communicate by message passing and also by accessing some shared memory (e.g., through some RDMA connections). We first consider the basic problem of implementing an atomic single-writer multi-reader (SWMR) register shared by all the processes in m&m systems. Specifically, we give an algorithm that implements such a register in m&m systems and show that it is optimal in the number of proc…
We explore the fundamental limits of distributed balls-into-bins algorithms. We present an adaptive symmetric algorithm that achieves a bin load of two in log* n+O(1) communication rounds using O(n) messages in total. Larger bin loads can be traded in for smaller time complexities. We prove a matching lower bound of (1-o(1))log* n on the time complexity of symmetric algorithms that guarantee small bin loads at an asymptotically optimal message complexity of O(n). For each assumption of the lower bound, we provide an algorithm violating it, in turn achieving a constant maximum bin load in const…
The Bulk-Synchronous Parallel model of computation has been used for the architecture independent design and analysis of parallel algorithms whose performance is expressed not only in terms of problem size n but also in terms of parallel machine properties. In this paper the performance of implementations of deterministic and randomized BSP sorting algorithms is examined. The deterministic algorithm uses deterministic regular oversampling and parallel sample sorting and is augmented to handle duplicate keys transparently with optimal asymptotic efficiency. The randomized algorithm is sample-so…
Most of today's high-speed switches and routers adopt an input-queued crossbar switch architecture. Such a switch needs to compute a matching (crossbar schedule) between the input ports and output ports during each switching cycle (time slot). A key research challenge in designing large (in number of input/output ports $N$) input-queued crossbar switches is to develop crossbar scheduling algorithms that can compute "high quality" matchings -- i.e., those that result in high switch throughput (ideally $100\%$) and low queueing delays for packets -- at line rates. SERENA is one such algorithm: i…
Designing and implementing efficient, provably correct parallel machine learning (ML) algorithms is challenging. Existing high-level parallel abstractions like MapReduce are insufficiently expressive while low-level tools like MPI and Pthreads leave ML experts repeatedly solving the same design challenges. By targeting common patterns in ML, we developed GraphLab, which improves upon abstractions like MapReduce by compactly expressing asynchronous iterative algorithms with sparse computational dependencies while ensuring data consistency and achieving a high degree of parallel performance. We …
Designing and implementing efficient, provably correct parallel machine learning (ML) algorithms is challenging. Existing high-level parallel abstractions like MapReduce are insufficiently expressive while low-level tools like MPI and Pthreads leave ML experts repeatedly solving the same design challenges. By targeting common patterns in ML, we developed GraphLab, which improves upon abstractions like MapReduce by compactly expressing asynchronous iterative algorithms with sparse computational dependencies while ensuring data consistency and achieving a high degree of parallel performance. We …
The MAX-MIN Ant System (MMAS) is one of the best-known Ant Colony Optimization (ACO) algorithms proven to be efficient at finding satisfactory solutions to many difficult combinatorial optimization problems. The slow-down in Moore's law, and the availability of graphics processing units (GPUs) capable of conducting general-purpose computations at high speed, has sparked considerable research efforts into the development of GPU-based ACO implementations. In this paper, we discuss a range of novel ideas for improving the GPU-based parallel MMAS implementation, allowing it to better utilize the c…
Parallelism has become extremely popular over the past decade, and there have been a lot of new parallel algorithms and software. The randomized work-stealing (RWS) scheduler plays a crucial role in this ecosystem. In this paper, we study two important topics related to the randomized work-stealing scheduler. Our first contribution is a simplified, classroom-ready version of analysis for the RWS scheduler. The theoretical efficiency of the RWS scheduler has been analyzed for a variety of settings, but most of them are quite complicated. In this paper, we show a new analysis, which we believe i…
We consider distributed optimization problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We develop unbiased parameter averaging methods for randomized second order optimization that employ sampling and sketching of the Hessian. Existing works do not take the bias of the estimators into consideration, which limits their application to massively parallel computation. We provide closed-form formulas for regularization parameters and step sizes that provably minimize the bias for sketched Newton directions. We also extend the framework …
In this paper we present randomized algorithms for sorting and convex hull that achieves optimal performance (for speed-up and cache misses) on the multicore model with private cache model. Our algorithms are cache oblivious and generalize the randomized divide and conquer strategy given by Reischuk and Reif and Sen. Although the approach yielded optimal speed-up in the PRAM model, we require additional techniques to optimize cache-misses in an oblivious setting. Under a mild assumption on input and number of processors our algorithm will have optimal time and cache misses with high probabilit…
A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity $O(\log D + \log \log_{\frac m n} n)$ and $O(m)$ space, for graphs on $n$ vertices with $m$ edges and diameter $D$. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present…
Progressive Hedging is a popular decomposition algorithm for solving multi-stage stochastic optimization problems. A computational bottleneck of this algorithm is that all scenario subproblems have to be solved at each iteration. In this paper, we introduce randomized versions of the Progressive Hedging algorithm able to produce new iterates as soon as a single scenario subproblem is solved. Building on the relation between Progressive Hedging and monotone operators, we leverage recent results on randomized fixed point methods to derive and analyze the proposed methods. Finally, we release the…
Maximum weight matching is one of the most fundamental combinatorial optimization problems with a wide range of applications in data mining and bioinformatics. Developing distributed weighted matching algorithms is challenging due to the sequential nature of efficient algorithms for this problem. In this paper, we develop a simple distributed algorithm for the problem on general graphs with approximation guarantee of $2+\varepsilon$ that (nearly) matches that of the sequential greedy algorithm. A key advantage of this algorithm is that it can be easily implemented in only two rounds of computa…
Collaborative filtering is amongst the most preferred techniques when implementing recommender systems. Recently, great interest has turned towards parallel and distributed implementations of collaborative filtering algorithms. This work is a survey of the parallel and distributed collaborative filtering implementations, aiming not only to provide a comprehensive presentation of the field's development, but also to offer future research orientation by highlighting the issues that need to be further developed.
In recent years there has been much interest in the Monte Carlo tree search algorithm, a new, adaptive, randomized optimization algorithm. In fields as diverse as Artificial Intelligence, Operations Research, and High Energy Physics, research has established that Monte Carlo tree search can find good solutions without domain dependent heuristics. However, practice shows that reaching high performance on large parallel machines is not so successful as expected. This paper proposes a new method for parallel Monte Carlo tree search based on the pipeline computation pattern.
Recently, the predicate detection problem was shown to be in the parallel complexity class NC. In this paper, we give the first work-optimal parallel algorithm to solve the predicate detection problem on a distributed computation with $n$ processes and at most $m$ states per process. The previous best known parallel predicate detection algorithm, ParallelCut, has time complexity $O(\log mn)$ and work complexity $O(m^3n^3\log mn)$. We give two algorithms, a deterministic algorithm with time complexity $O(mn)$ and work complexity $O(mn^2)$, and a randomized algorithm with time complexity $(mn)^{…
This paper analyzes the cache miss cost of algorithms when scheduled using randomized work stealing (RWS) in a parallel environment, taking into account the effects of false sharing. First, prior analyses (due to Acar et al.) are extended to incorporate false sharing. However, to control the possible delays due to false sharing, some restrictions on the algorithms seem necessary. Accordingly, the class of Hierarchical Tree algorithms is introduced and their performance analyzed. In addition, the paper analyzes the performance of a subclass of the Hierarchical Tree Algorithms, called HBP algori…
The disjoint set union problem is a basic problem in data structures with a wide variety of applications. We extend a known efficient sequential algorithm for this problem to obtain a simple and efficient concurrent wait-free algorithm running on an asynchronous parallel random access machine (APRAM). Crucial to our result is the use of randomization. Under a certain independence assumption, for a problem instance in which there are n elements, m operations, and p processes, our algorithm does Theta(m (alpha(n, m/(np)) + log(np/m + 1))) expected work, where the expectation is over the random c…
We study a $(1+ε)$-approximate single-source shortest paths (henceforth, $(1+ε)$-SSSP) in $n$-vertex undirected, weighted graphs in the parallel (PRAM) model of computation. A randomized algorithm with polylogarithmic time and slightly super-linear work $\tilde{O}(|E|\cdot n^ρ)$, for an arbitrarily small $ρ>0$, was given by Cohen [Coh94] more than $25$ years ago. Exciting progress on this problem was achieved in recent years [ElkinN17,ElkinN19,Li19,AndoniSZ19], culminating in randomized polylogarithmic time and $\tilde{O}(|E|)$ work. However, the question of whether there exists a deterministi…
University of Torino（都灵大学）；Noesis Solutions NV（Noesis Solutions NV公司）；Pacific Northwest National Lab.（太平洋西北国家实验室）
This work aims to assess the state of the art of data parallel deep neural network training, trying to identify potential research tracks to be exploited for performance improvement. Beside, it presents a design for a practical C++ library dedicated at implementing and unifying the current state of the art methodologies for parallel training in a performance-conscious framework, allowing the user to explore novel strategies without departing significantly from its usual work-flow.
We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems. We derive novel approximation guarantees for classical sketching methods and establish tight concentration results that serve as both upper and lower bounds on the error. We then extend our analysis to the accuracy of parameter averaging for distributed sketches. Furt…
A future is a programming construct designed for concurrent and asynchronous evaluation of code, making it particularly useful for parallel processing. The future package implements the Future API for programming with futures in R. This minimal API provides sufficient constructs for implementing parallel versions of well-established, high-level map-reduce APIs. The future ecosystem supports exception handling, output and condition relaying, parallel random number generation, and automatic identification of globals lowering the threshold to parallelize code. The Future API bridges parallel fron…
We present Matrix Distributed Processing, a C++ library for fast development of efficient parallel algorithms. MDP is based on MPI and consists of a collection of C++ classes and functions such as lattice, site and field. Once an algorithm is written using these components the algorithm is automatically parallel and no explicit call to communication functions is required. MDP is particularly suitable for implementing parallel solvers for multi-dimensional differential equations and mesh-like problems.
University of Nantes（南特大学）；University of Tokyo（东京大学）；UPMC（巴黎第六大学）
We propose a probabilistic model for the parallel execution of Las Vegas algorithms, i.e., randomized algorithms whose runtime might vary from one execution to another, even with the same input. This model aims at predicting the parallel performances (i.e., speedups) by analysis the runtime distribution of the sequential runs of the algorithm. Then, we study in practice the case of a particular Las Vegas algorithm for combinatorial optimization, on three classical problems, and compare with an actual parallel implementation up to 256 cores. We show that the prediction can be quite accurate, ma…
