Stochastic variational inference (SVI) employs stochastic optimization to scale up Bayesian computation to massive data. Since SVI is at its core a stochastic gradient-based algorithm, horizontal parallelism can be harnessed to allow larger scale inference. We propose a lock-free parallel implementation for SVI which allows distributed computations over multiple slaves in an asynchronous style. We show that our implementation leads to linear speed-up while guaranteeing an asymptotic ergodic convergence rate @math ) given that the number of slaves is bounded by @math ( @math is the total number of iterations). The implementation is done in a high-performance computing (HPC) environment using message passing interface (MPI) for python (MPI4py). The extensive empirical evaluation shows that our parallel SVI is lossless, performing comparably well to its counterpart serial SVI with linear speed-up.
Few work has been proposed to scale variational inference to large datasets. We can distinguish two major classes. The first class is based on the Bayesian filtering approach @cite_2 @cite_14 . That is, the sequential nature of Bayes theorem is exploited to recursively update an approximation of the posterior. Particularly, variational inference is used between the updates to approximate the posterior which becomes the prior of the next step. Author in @cite_2 employs forgetting factors to decay the contributions from old data points in favour of a new better one. The algorithm proposed in @cite_14 considers a sequence of data batches. It iterates over the data points in the batch until convergence. The computation of the batches posterior is done in a distributed and asynchronous manner. That is, the algorithm applies VI by performing asynchronous Bayesian updates to the posterior as batches of data arrive continuously. Similar to our approach, master-slave architecture is used.
@cite_14 @cite_2
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We provide a general framework for privacy-preserving variational Bayes (VB) for a large class of probabilistic models, called the conjugate exponential (CE) family. Our primary observation is that when models are in the CE family, we can privatise the variational posterior distributions simply by perturbing the expected sufficient statistics of the complete-data likelihood. For widely used non-CE models with binomial likelihoods, we exploit the P 'o lya-Gamma data augmentation scheme to bring such models into the CE family, such that inferences in the modified model resemble the private variational Bayes algorithm as closely as possible. The iterative nature of variational Bayes presents a further challenge since iterations increase the amount of noise needed. We overcome this by combining: (1) a relaxed notion of differential privacy, called concentrated differential privacy, which provides a tight bound on the privacy cost of multiple VB iterations and thus significantly decreases the amount of additive noise; and (2) the privacy amplification effect of subsampling mini-batches from large-scale data in stochastic learning. We empirically demonstrate the effectiveness of our method in CE and non-CE models including latent Dirichlet allocation, Bayesian logistic regression, and sigmoid belief networks, evaluated on real-world datasets. Variational Inference is a popular technique to approximate a possibly intractable Bayesian posterior with a more tractable one. Recently, Boosting Variational Inference has been proposed as a new paradigm to approximate the posterior by a mixture of densities by greedily adding components to the mixture. In the present work, we study the convergence properties of this approach from a modern optimization viewpoint by establishing connections to the classic Frank-Wolfe algorithm. Our analyses yields novel theoretical insights on the Boosting of Variational Inference regarding the sufficient conditions for convergence, explicit sublinear linear rates, and algorithmic simplifications. This paper presents a methodology for creating streaming, distributed inference algorithms for Bayesian nonparametric (BNP) models. In the proposed framework, processing nodes receive a sequence of data minibatches, compute a variational posterior for each, and make asynchronous streaming updates to a central model. In contrast to previous algorithms, the proposed framework is truly streaming, distributed, asynchronous, learning-rate-free, and truncation-free. The key challenge in developing the framework, arising from the fact that BNP models do not impose an inherent ordering on their components, is finding the correspondence between minibatch and central BNP posterior components before performing each update. To address this, the paper develops a combinatorial optimization problem over component correspondences, and provides an efficient solution technique. The paper concludes with an application of the methodology to the DP mixture model, with experimental results demonstrating its practical scalability and performance. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.