Communication costs, resulting from synchronization requirements during learning, can greatly slow down many parallel machine learning algorithms. In this paper, we present a parallel Markov chain Monte Carlo (MCMC) algorithm in which subsets of data are processed independently, with very little communication. First, we arbitrarily partition data onto multiple machines. Then, on each machine, any classical MCMC method (e.g., Gibbs sampling) may be used to draw samples from a posterior distribution given the data subset. Finally, the samples from each machine are combined to form samples from the full posterior. This embarrassingly parallel algorithm allows each machine to act independently on a subset of the data (without communication) until the final combination stage. We prove that our algorithm generates asymptotically exact samples and empirically demonstrate its ability to parallelize burn-in and sampling in several models.
Several parallel MCMC algorithms have been designed for specific models, such as for topic models @cite_6 @cite_15 and nonparametric mixture models @cite_14 . These approaches still require synchronization to be correct (or approximately correct), while ours aims for more general model settings and does not need synchronization until the final combination stage.
@cite_15 @cite_14 @cite_6
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Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. Nonparametric mixture models based on the Dirichlet process are an elegant alternative to finite models when the number of underlying components is unknown, but inference in such models can be slow. Existing attempts to parallelize inference in such models have relied on introducing approximations, which can lead to inaccuracies in the posterior estimate. In this paper, we describe auxiliary variable representations for the Dirichlet process and the hierarchical Dirichlet process that allow us to perform MCMC using the correct equilibrium distribution, in a distributed manner. We show that our approach allows scalable inference without the deterioration in estimate quality that accompanies existing methods. Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean square error (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of L-4 5 at L iterations, compared to L-2 3 for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications. S U M M A R Y Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It’s generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences. We investigate the use of adaptive MCMC algorithms to automatically tune the Markov chain parameters during a run. Examples include the Adaptive Metropolis (AM) multivariate algorithm of Haario, Saksman, and Tamminen (2001), Metropolis-within-Gibbs algorithms for nonconjugate hierarchical models, regionally adjusted Metropolis algorithms, and logarithmic scalings. Computer simulations indicate that the algorithms perform very well compared to nonadaptive algorithms, even in high dimension. We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions. We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Markov chains, a novel family of Markov chains leveraging model symmetries to reduce mixing times. We establish an insightful connection between model symmetries and rapid mixing of orbital Markov chains. Thus, we present the first lifted MCMC algorithm for probabilistic graphical models. Both analytical and empirical results demonstrate the effectiveness and efficiency of the approach. Tasks such as record linkage and multi-target tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carlo (MCMC) techniques with customized proposal distributions. Currently, implementing such a system requires coding MCMC state representations and acceptance probability calculations that are specific to a particular application. An alternative approach, which we pursue in this paper, is to use a general-purpose probabilistic modeling language (such as BLOG) and a generic Metropolis-Hastings MCMC algorithm that supports user-supplied proposal distributions. Our algorithm gains flexibility by using MCMC states that are only partial descriptions of possible worlds; we provide conditions under which MCMC over partial worlds yields correct answers to queries. We also show how to use a context-specific Bayes net to identify the factors in the acceptance probability that need to be computed for a given proposed move. Experimental results on a citation matching task show that our general-purpose MCMC engine compares favorably with an application-specific system. Stochastic gradient Markov chain Monte Carlo (SG-MCMC) methods are Bayesian analogs to popular stochastic optimization methods; however, this connection is not well studied. We explore this relationship by applying simulated annealing to an SGMCMC algorithm. Furthermore, we extend recent SG-MCMC methods with two key components: i) adaptive preconditioners (as in ADAgrad or RMSprop), and ii) adaptive element-wise momentum weights. The zerotemperature limit gives a novel stochastic optimization method with adaptive elementwise momentum weights, while conventional optimization methods only have a shared, static momentum weight. Under certain assumptions, our theoretical analysis suggests the proposed simulated annealing approach converges close to the global optima. Experiments on several deep neural network models show state-of-the-art results compared to related stochastic optimization algorithms. In 1999 Kannan, Tet ali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklos, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklos, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. Dirichlet process (DP) mixture models are the cornerstone of non- parametric Bayesian statistics, and the development of Monte-Carlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of non- parametric Bayesian methods to a variety of practical data analysis problems. However, MCMC sampling can be prohibitively slow, and it is important to ex- plore alternatives. One class of alternatives is provided by variational methods, a class of deterministic algorithms that convert inference problems into optimization problems (Opper and Saad 2001; Wainwright and Jordan 2003). Thus far, varia- tional methods have mainly been explored in the parametric setting, in particular within the formalism of the exponential family (Attias 2000; Ghahramani and Beal 2001; 2003). In this paper, we present a variational inference algorithm for DP mixtures. We present experiments that compare the algorithm to Gibbs sampling algorithms for DP mixtures of Gaussians and present an application to a large-scale image analysis problem. The Markov Chain Monte Carlo (MCMC) family of methods form a valuable part of the toolbox of social modeling and prediction techniques, enabling modelers to generate samples and summary statistics of a population of interest with minimal information. It has been used successfully to model changes over time in many types of social systems, including patterns of disease spread, adolescent smoking, and geopolitical conflicts. In MCMC an initial proposal distribution is iteratively refined until it approximates the posterior distribution. However, the selection of the proposal distribution can have a significant impact on model convergence. In this paper, we propose a new hybrid modeling technique in which an agent-based model is used to initialize the proposal distribution of the MCMC simulation. We demonstrate the use of our modeling technique in an urban transportation prediction scenario and show that the hybrid combined model produces more accurate predictions than either of the parent models. Abstract This article describes a new Metropolis-like transition rule, the multiple-try Metropolis, for Markov chain Monte Carlo (MCMC) simulations. By using this transition rule together with adaptive direction sampling, we propose a novel method for incorporating local optimization steps into a MCMC sampler in continuous state-space. Numerical studies show that the new method performs significantly better than the traditional Metropolis-Hastings (M-H) sampler. With minor tailoring in using the rule, the multiple-try method can also be exploited to achieve the effect of a griddy Gibbs sampler without having to bear with griddy approximations, and the effect of a hit-and-run algorithm without having to figure out the required conditional distribution in a random direction. In human-robot collaborative tasks, incorporating qualitative information provided by humans can greatly enhance the robustness and efficacy of robot state estimation. We introduce an algorithmic framework to model qualitative information as quantitative constraints on and between states. Our approach, named Sequentially Constrained Hamiltonian Monte Carlo, integrates Hamiltonian dynamics into Sequentially Constrained Monte Carlo sampling. We are able to generate samples that satisfy arbitrarily complex, non-smooth and discontinuous constraints, which in turn allows us to support a wide range of qualitative information. We evaluate our approach for constrained sampling qualitatively and quantitatively with several classes of constraints. SCHMC significantly outperforms the Metropolis-Hastings algorithm (a standard Markov Chain Monte Carlo (MCMC) method) and the Hamiltonian Monte Carlo (HMC) method, in terms of both the accuracy of the sampling (for satisfying constraints) and the quality of approximation. Compared to Sequentially Constrained Monte Carlo (SCMC), which supports similar kinds of constraints, our SCHMC approach has faster convergence rates and lower parameter sensitivity. The parallelization of MCTS across multiple-machines has proven surprisingly difficult. The limitations of existing algorithms were evident in the 2009 Computer Olympiad where ZEN using a single fourcore machine defeated both Fuego with ten eight-core machines, and Mogo with twenty thirty-two core machines. This paper investigates the limits of parallel MCTS in order to understand why distributed parallelism has proven so difficult and to pave the way towards future distributed algorithms with better scaling. We first analyze the single-threaded scaling of Fuego and find that there is an upper bound on the play-quality improvements which can come from additional search. We then analyze the scaling of an idealized N-core shared memory machine to determine the maximum amount of parallelism supported by MCTS. We show that parallel speedup depends critically on how much time is given to each player. We use this relationship to predict parallel scaling for time scales beyond what can be empirically evaluated due to the immense computation required. Our results show that MCTS can scale nearly perfectly to at least 64 threads when combined with virtual loss, but without virtual loss scaling is limited to just eight threads. We also find that for competition time controls scaling to thousands of threads is impossible not necessarily due to MCTS not scaling, but because high levels of parallelism can start to bump up against the upper performance bound of FUEGO itself. Many scientific applications are programmed using hybrid programming models that use both message passing and shared memory, due to the increasing prevalence of large-scale systems with multicore, multisocket nodes. Previous work has shown that energy efficiency can be improved using software-controlled execution schemes that consider both the programming model and the power-aware execution capabilities of the system. However, such approaches have focused on identifying optimal resource utilization for one programming model, either shared memory or message passing, in isolation. The potential solution space, thus the challenge, increases substantially when optimizing hybrid models since the possible resource configurations increase exponentially. Nonetheless, with the accelerating adoption of hybrid programming models, we increasingly need improved energy efficiency in hybrid parallel applications on large-scale systems. In this work, we present new software-controlled execution schemes that consider the effects of dynamic concurrency throttling (DCT) and dynamic voltage and frequency scaling (DVFS) in the context of hybrid programming models. Specifically, we present predictive models and novel algorithms based on statistical analysis that anticipate application power and time requirements under different concurrency and frequency configurations. We apply our models and methods to the NPB MZ benchmarks and selected applications from the ASC Sequoia codes. Overall, we achieve substantial energy savings (8.74 percent on average and up to 13.8 percent) with some performance gain (up to 7.5 percent) or negligible performance loss. In this paper, we design and analyze parallel algorithms for skyline queries. The skyline of a multidimensional set consists of the points for which no other point exists that is at least as good along every dimension. As a framework for parallel computation, we use both the MP model proposed in Koutris and Suciu (2011), which requires that the data is perfectly load-balanced, and a variation of the model in Afrati and Ullman (2010), the GMP model, which demands weaker load balancing constraints. In addition to load balancing, we want to minimize the number of blocking steps, where all processors must wait and synchronize. We propose a 2-step algorithm in the MP model for any dimension of the dataset, as well a 1-step algorithm for the case of 2 and 3 dimensions. Finally, we present a 1-step algorithm in the GMP model for any number of dimensions and a 1-step algorithm in the MP model for uniform distributions of data points. Despite having various attractive qualities such as high prediction accuracy and the ability to quantify uncertainty and avoid over-fitting, Bayesian Matrix Factorization has not been widely adopted because of the prohibitive cost of inference. In this paper, we propose a scalable distributed Bayesian matrix factorization algorithm using stochastic gradient MCMC. Our algorithm, based on Distributed Stochastic Gradient Langevin Dynamics, can not only match the prediction accuracy of standard MCMC methods like Gibbs sampling, but at the same time is as fast and simple as stochastic gradient descent. In our experiments, we show that our algorithm can achieve the same level of prediction accuracy as Gibbs sampling an order of magnitude faster. We also show that our method reduces the prediction error as fast as distributed stochastic gradient descent, achieving a 4.1 improvement in RMSE for the Netflix dataset and an 1.8 for the Yahoo music dataset. The field of Distributed Constraint Optimization (DCOP) has gained momentum in recent years, thanks to its ability to address various applications related to multi-agent coordination. Nevertheless, solving DCOPs is computationally challenging. Thus, in large scale, complex applications, incomplete DCOP algorithms are necessary. Recently, researchers have introduced a promising class of incomplete DCOP algorithms, based on sampling. However, this paradigm requires a multitude of samples to ensure convergence. This paper exploits the property that sampling is amenable to parallelization, and introduces a general framework, called Distributed MCMC (DMCMC), that is based on a dynamic programming procedure and uses Markov Chain Monte Carlo (MCMC) sampling algorithms to solve DCOPs. Additionally, DMCMC harnesses the parallel computing power of Graphical Processing Units (GPUs) to speed-up the sampling process. The experimental results show that DMCMC can find good solutions up to two order of magnitude faster than other incomplete DCOP algorithms. Multicell coordinated beamforming (MCBF), where multiple base stations (BSs) collaborate with each other in the beamforming design for mitigating the intercell interference (ICI), has been a subject drawing great attention recently. Most MCBF designs assume perfect channel state information (CSI) of mobile stations (MSs); however CSI errors are inevitable at the BSs in practice. Assuming elliptically bounded CSI errors, this paper studies the robust MCBF design problem that minimizes the weighted sum power of BSs subject to worst-case signal-to-interference-plus-noise ratio (SINR) constraints on the MSs. Our goal is to devise a distributed optimization method to obtain the worst-case robust beamforming solutions in a decentralized fashion with only local CSI used at each BS and limited backhaul information exchange between BSs. However, the considered problem is difficult to handle even in the centralized form. We first propose an efficient approximation method for solving the nonconvex centralized problem, using semidefinite relaxation (SDR), an approximation technique based on convex optimization. Then a distributed robust MCBF algorithm is further proposed, using a distributed convex optimization technique known as alternating direction method of multipliers (ADMM). We analytically show the convergence of the proposed distributed robust MCBF algorithm to the optimal centralized solution. We also extend the worst-case robust beamforming design as well as its decentralized implementation method to a fully coordinated scenario. Simulation results are presented to examine the effectiveness of the proposed SDR method and the distributed robust MCBF algorithm.