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.
Consensus Monte Carlo @cite_12 is perhaps the most relevant work to ours. In this algorithm, data is also portioned into different machines and MCMC is performed independently on each machine. Thus, it roughly has the same time complexity as our algorithm. However, the prior is not explicitly reweighted during sampling as we do in Eq , and final samples for the full posterior are generated by averaging subposterior samples. Furthermore, this algorithm has few theoretical guarantees. We find that this algorithm can be viewed as a relaxation of our nonparametric, asymptotically exact sampling procedure, where samples are generated from an evenly weighted mixture (instead of each component having weight @math ) and where each sample is set to @math instead of being drawn from @math . This algorithm is one of our experimental baselines.
@cite_12
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The class of species sampling mixture models is introduced as an exten- sion of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior par- tition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirich- let process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Polya urn Gibbs sampling and a Polya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparamet- ric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors. Importance sampling methods can be iterated like MCMC algorithms, while being more robust against dependence and starting values. The population Monte Carlo principle consists of iterated generations of importance samples, with importance functions depending on the previously generated importance samples. The advantage over MCMC algorithms is that the scheme is unbiased at any iteration and can thus be stopped at any time, while iterations improve the performances of the importance function, thus leading to an adaptive importance sampling. We illustrate this method on a mixture example with multiscale importance functions. A second example reanalyzes the ion channel model using an importance sampling scheme based on a hidden Markov representation, and compares population Monte Carlo with a corresponding MCMC algorithm. Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any @math @math -dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses @math samples and runs in time @math , both significantly lower than previously known. The constant factor @math is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that @math samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses @math samples and runs in time @math . Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the @math distance to an unknown underlying distribution. A @math -modal probability distribution over the discrete domain @math is one whose histogram has at most @math "peaks" and "valleys." Such distributions are natural generalizations of monotone ( @math ) and unimodal ( @math ) probability distributions, which have been intensively studied in probability theory and statistics. In this paper we consider the problem of (i.e., performing density estimation of) an unknown @math -modal distribution with respect to the @math distance. The learning algorithm is given access to independent samples drawn from an unknown @math -modal distribution @math , and it must output a hypothesis distribution @math such that with high probability the total variation distance between @math and @math is at most @math Our main goal is to obtain algorithms for this problem that use (close to) an information-theoretically optimal number of samples. We give an efficient algorithm for this problem that runs in time @math . For @math , the number of samples used by our algorithm is very close (within an @math factor) to being information-theoretically optimal. Prior to this work computationally efficient algorithms were known only for the cases @math Birge:87b,Birge:97 . A novel feature of our approach is that our learning algorithm crucially uses a new algorithm for as a key subroutine. The learning algorithm uses the property tester to efficiently decompose the @math -modal distribution into @math (near-)monotone distributions, which are easier to learn. We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on @math , for all @math . Prior to this work, no finite sample upper bound was known for this estimator in more than @math dimensions. In more detail, we prove that for any @math and @math , given @math samples drawn from an unknown log-concave density @math on @math , the MLE outputs a hypothesis @math that with high probability is @math -close to @math , in squared Hellinger loss. A sample complexity lower bound of @math was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an @math factor. Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have dimension-independent bounds for the underlying @math processes under sufficiently strong smoothness conditions, the resulting discrete algorithms have complexity and number of function evaluations growing with the dimension. Motivated by this problem, in this paper, we give a general algorithm for solving multivariate ordinary differential equations whose solution is close to the span of a known basis of functions (e.g., polynomials or piecewise polynomials). The resulting algorithm has polylogarithmic depth and essentially tight runtime - it is nearly linear in the size of the representation of the solution. We apply this to the sampling problem to obtain a nearly linear implementation of HMC for a broad class of smooth, strongly logconcave densities, with the number of iterations (parallel depth) and gradient evaluations being @math in the dimension (rather than polynomial as in previous work). This class includes the widely-used loss function for logistic regression with incoherent weight matrices and has been subject of much study recently. We also give a faster algorithm with @math for the more general and standard class of strongly convex functions with Lipschitz gradient. These results are based on (1) an improved contraction bound for the exact HMC process and (2) logarithmic bounds on the degree of polynomials that approximate solutions of the differential equations arising in implementing HMC. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning @math (where @math is a bound on the smoothness and @math on the strong convexity) to a linear dependence on @math . Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate. We give algorithms with provable guarantees that learn a class of deep nets in the generative model view popularized by Hinton and others. Our generative model is an @math node multilayer neural net that has degree at most @math for some @math and each edge has a random edge weight in @math . Our algorithm learns almost all networks in this class with polynomial running time. The sample complexity is quadratic or cubic depending upon the details of the model. The algorithm uses layerwise learning. It is based upon a novel idea of observing correlations among features and using these to infer the underlying edge structure via a global graph recovery procedure. The analysis of the algorithm reveals interesting structure of neural networks with random edge weights. Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures, especially in the high-dimensional problems. This paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on @math , for some integer @math . We establish nonasymptotic bounds for the error of approximating the true distribution by the one obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous-time diffusion processes, which may be of interest beyond the framework of distributions with log-concave densities considered in the present work. We present a new sampling approach to Bayesian learning of the Bayesian network structure. Like some earlier sampling methods, we sample linear orders on nodes rather than directed acyclic graphs (DAGs). The key difference is that we replace the usual Markov chain Monte Carlo (MCMC) method by the method of annealed importance sampling (AIS). We show that AIS is not only competitive to MCMC in exploring the posterior, but also superior to MCMC in two ways: it enables easy and efficient parallelization, due to the independence of the samples, and lower-bounding of the marginal likelihood of the model with good probabilistic guarantees. We also provide a principled way to correct the bias due to order-based sampling, by implementing a fast algorithm for counting the linear extensions of a given partial order. Given @math elements with nonnegative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error @math in time polynomial in @math and @math (fully polynomial approximation scheme). More precisely, our algorithm takes time @math . Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques and subsequently by Dyer via dynamic programming and rejection sampling. Many machine learning tasks, such as learning with invariance and policy evaluation in reinforcement learning, can be characterized as problems of learning from conditional distributions. In such problems, each sample @math itself is associated with a conditional distribution @math represented by samples @math , and the goal is to learn a function @math that links these conditional distributions to target values @math . These learning problems become very challenging when we only have limited samples or in the extreme case only one sample from each conditional distribution. Commonly used approaches either assume that @math is independent of @math , or require an overwhelmingly large samples from each conditional distribution. To address these challenges, we propose a novel approach which employs a new min-max reformulation of the learning from conditional distribution problem. With such new reformulation, we only need to deal with the joint distribution @math . We also design an efficient learning algorithm, Embedding-SGD, and establish theoretical sample complexity for such problems. Finally, our numerical experiments on both synthetic and real-world datasets show that the proposed approach can significantly improve over the existing algorithms. We consider streaming, one-pass principal component analysis (PCA), in the high-dimensional regime, with limited memory. Here, @math -dimensional samples are presented sequentially, and the goal is to produce the @math -dimensional subspace that best approximates these points. Standard algorithms require @math memory; meanwhile no algorithm can do better than @math memory, since this is what the output itself requires. Memory (or storage) complexity is most meaningful when understood in the context of computational and sample complexity. Sample complexity for high-dimensional PCA is typically studied in the setting of the spiked covariance model , where @math -dimensional points are generated from a population covariance equal to the identity (white noise) plus a low-dimensional perturbation (the spike) which is the signal to be recovered. It is now well-understood that the spike can be recovered when the number of samples, @math , scales proportionally with the dimension, @math . Yet, all algorithms that provably achieve this, have memory complexity @math . Meanwhile, algorithms with memory-complexity @math do not have provable bounds on sample complexity comparable to @math . We present an algorithm that achieves both: it uses @math memory (meaning storage of any kind) and is able to compute the @math -dimensional spike with @math sample-complexity -- the first algorithm of its kind. While our theoretical analysis focuses on the spiked covariance model, our simulations show that our algorithm is successful on much more general models for the data. We revisit a problem introduced by Bharat and Broder almost a decade ago: how to sample random pages from the corpus of documents indexed by a search engine, using only the search engine’s public interface? Such a primitive is particularly useful in creating objective benchmarks for search engines. The technique of Bharat and Broder suffers from a well-recorded bias: it favors long documents. In this paper we introduce two novel sampling algorithms: a lexicon-based algorithm and a random walk algorithm. Our algorithms produce biased samples, but each sample is accompanied by a weight, which represents its bias. The samples, in conjunction with the weights, are then used to simulate near-uniform samples. To this end, we resort to four well-known Monte Carlo simulation methods: rejection sampling, importance sampling, the Metropolis-Hastings algorithm, and the Maximum Degree method. The limited access to search engines force our algorithms to use bias weights that are only “approximate”. We characterize analytically the effect of approximate bias weights on Monte Carlo methods and conclude that our algorithms are guaranteed to produce near-uniform samples from the search engine’s corpus. Our study of approximate Monte Carlo methods could be of independent interest. Experiments on a corpus of 2.4 million documents substantiate our analytical findings and show that our algorithms do not have significant bias towards long documents. We use our algorithms to collect fresh comparative statistics about the corpora of the Google, MSN Search, and Yahoo! search engines. We study the asymptotic behavior of the persistent homology of i.i.d. samples from a @math -Ahlfors regular measure --- one that satisfies uniform bounds of the form for some @math all @math in the support of @math and all sufficiently small @math Our main result is that if @math are sampled from a @math -Ahlfors regular measure on @math and @math denotes the @math -weight of the minimal spanning tree on @math [E_ (x_1, ,x_n )= e T (x_1, ,x_n ) |e|^ ] then [E_ (x_1, ,x_n ) n^ d- d ] with high probability as @math We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology. As an application, we exhibit hypotheses under which the fractal dimension of a measure can be computed from the persistent homology of i.i.d. samples from that space, in a manner similar to that proposed in the experimental work of (2018). In sparse recovery we are given a matrix @math (the dictionary) and a vector of the form @math where @math is sparse, and the goal is to recover @math . This is a central notion in signal processing, statistics and machine learning. But in applications such as sparse coding, edge detection, compression and super resolution, the dictionary @math is unknown and has to be learned from random examples of the form @math where @math is drawn from an appropriate distribution --- this is the dictionary learning problem. In most settings, @math is overcomplete: it has more columns than rows. This paper presents a polynomial-time algorithm for learning overcomplete dictionaries; the only previously known algorithm with provable guarantees is the recent work of Spielman, Wang and Wright who gave an algorithm for the full-rank case, which is rarely the case in applications. Our algorithm applies to incoherent dictionaries which have been a central object of study since they were introduced in seminal work of Donoho and Huo. In particular, a dictionary is @math -incoherent if each pair of columns has inner product at most @math . The algorithm makes natural stochastic assumptions about the unknown sparse vector @math , which can contain @math non-zero entries (for any @math ). This is close to the best @math allowable by the best sparse recovery algorithms even if one knows the dictionary @math exactly. Moreover, both the running time and sample complexity depend on @math , where @math is the target accuracy, and so our algorithms converge very quickly to the true dictionary. Our algorithm can also tolerate substantial amounts of noise provided it is incoherent with respect to the dictionary (e.g., Gaussian). In the noisy setting, our running time and sample complexity depend polynomially on @math , and this is necessary. In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (, 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class @math (or @math ) and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence we prove the following bounds: 1. @math , where @math is the Littlestone's (1987) dimension characterizing the number of mistakes in the online-mistake-bound learning model. Known bounds on @math then imply that @math can be much higher than the VC-dimension of @math . 2. For any @math , there exists a class @math such that @math but @math . 3. For any @math , there exists a class @math such that the sample complexity of (pure) @math -differentially private PAC learning is @math but the sample complexity of the relaxed @math -differentially private PAC learning is @math . This resolves an open problem of (2013b). In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order @math for all pairs @math , where @math is a constant and @math for all @math and @math . It is assumed that the errors are independent. Given the status of the queries the goal is to find the maximum likelihood order. In other words, the goal is find a permutation @math that minimizes the number of pairs @math where @math . The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbations of a linear order. Note that when @math and @math is large, it is impossible to recover the original order @math . It is known that the weighted feedback are set problem on tournaments is NP-hard in general. Here we present an algorithm of running time @math and sampling complexity @math that with high probability solves the noisy sorting without re-sampling problem. We also show that if @math is an optimal solution of the problem then it is close'' to the original order. More formally, with high probability it holds that @math and @math . Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts. 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. We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ λ c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.