Summary Stochastic gradient Markov chain Monte Carlo algorithms have received much attention in Bayesian computing for big data problems, but they are only applicable to a small class of problems for which the parameter space has a fixed dimension and the log-posterior density is differentiable with respect to the parameters. This paper proposes an extended stochastic gradient Markov chain Monte Carlo algorithm which, by introducing appropriate latent variables, can be applied to more general large-scale Bayesian computing problems, such as those involving dimension jumping and missing data. Numerical studies show that the proposed algorithm is highly scalable and much more efficient than traditional Markov chain Monte Carlo algorithms.
more »
« less
Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo
Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte
Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for
Bayesian inference that can scale to large datasets, allowing to sample from the posterior
distribution of the parameters of a statistical model given the input data and the prior
distribution over the model parameters. However, these algorithms do not apply to the
decentralized learning setting, when a network of agents are working collaboratively to learn
the parameters of a statistical model without sharing their individual data due to privacy
reasons or communication constraints. We study two algorithms: Decentralized SGLD
(DE-SGLD) and Decentralized SGHMC (DE-SGHMC) which are adaptations of SGLD
and SGHMC methods that allow scaleable Bayesian inference in the decentralized setting
for large datasets. We show that when the posterior distribution is strongly log-concave and
smooth, the iterates of these algorithms converge linearly to a neighborhood of the target
distribution in the 2-Wasserstein distance if their parameters are selected appropriately.
We illustrate the efficiency of our algorithms on decentralized Bayesian linear regression
and Bayesian logistic regression problems
more »
« less
- NSF-PAR ID:
- 10326387
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 22
- ISSN:
- 1532-4435
- Page Range / eLocation ID:
- 1-69
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
We investigate solution methods for large-scale inverse problems governed by partial differential equations (PDEs) via Bayesian inference. The Bayesian framework provides a statistical setting to infer uncertain parameters from noisy measurements. To quantify posterior uncertainty, we adopt Markov Chain Monte Carlo (MCMC) approaches for generating samples. To increase the efficiency of these approaches in high-dimension, we make use of local information about gradient and Hessian of the target potential, also via Hamiltonian Monte Carlo (HMC). Our target application is inferring the field of soil permeability processing observations of pore pressure, using a nonlinear PDE poromechanics model for predicting pressure from permeability. We compare the performance of different sampling approaches in this and other settings. We also investigate the effect of dimensionality and non-gaussianity of distributions on the performance of different sampling methods.
-
null (Ed.)Stochastic Gradient Langevin Dynamics (SGLD) have been widely used for Bayesian sampling from certain probability distributions, incorporating derivatives of the log-posterior. With the derivative evaluation of the log-posterior distribution, SGLD methods generate samples from the distribution through performing as a thermostats dynamics that traverses over gradient flows of the log-posterior with certainly controllable perturbation. Even when the density is not known, existing solutions still can first learn the kernel density models from the given datasets, then produce new samples using the SGLD over the kernel density derivatives. In this work, instead of exploring new samples from kernel spaces, a novel SGLD sampler, namely, Randomized Measurement Langevin Dynamics (RMLD) is proposed to sample the high-dimensional sparse representations from the spectral domain of a given dataset. Specifically, given a random measurement matrix for sparse coding, RMLD first derives a novel likelihood evaluator of the probability distribution from the loss function of LASSO, then samples from the high-dimensional distribution using stochastic Langevin dynamics with derivatives of the logarithm likelihood and Metropolis–Hastings sampling. In addition, new samples in low-dimensional measuring spaces can be regenerated using the sampled high-dimensional vectors and the measurement matrix. The algorithm analysis shows that RMLD indeed projects a given dataset into a high-dimensional Gaussian distribution with Laplacian prior, then draw new sparse representation from the dataset through performing SGLD over the distribution. Extensive experiments have been conducted to evaluate the proposed algorithm using real-world datasets. The performance comparisons on three real-world applications demonstrate the superior performance of RMLD beyond baseline methods.more » « less
-
Meila, Marina ; Zhang, Tong (Ed.)Black-box variational inference algorithms use stochastic sampling to analyze diverse statistical models, like those expressed in probabilistic programming languages, without model-specific derivations. While the popular score-function estimator computes unbiased gradient estimates, its variance is often unacceptably large, especially in models with discrete latent variables. We propose a stochastic natural gradient estimator that is as broadly applicable and unbiased, but improves efficiency by exploiting the curvature of the variational bound, and provably reduces variance by marginalizing discrete latent variables. Our marginalized stochastic natural gradients have intriguing connections to classic coordinate ascent variational inference, but allow parallel updates of variational parameters, and provide superior convergence guarantees relative to naive Monte Carlo approximations. We integrate our method with the probabilistic programming language Pyro and evaluate real-world models of documents, images, networks, and crowd-sourcing. Compared to score-function estimators, we require far fewer Monte Carlo samples and consistently convergence orders of magnitude faster.more » « less
-
We propose a stochastic variational inference algorithm for training large-scale Bayesian networks, where noisy-OR conditional distributions are used to capture higher-order relationships. One application is to the learning of hierarchical topic models for text data. While previous work has focused on two-layer networks popular in applications like medical diagnosis, we develop scalable algorithms for deep networks that capture a multi-level hierarchy of interactions. Our key innovation is a family of constrained variational bounds that only explicitly optimize posterior probabilities for the sub-graph of topics most related to the sparse observations in a given document. These constrained bounds have comparable accuracy but dramatically reduced computational cost. Using stochastic gradient updates based on our variational bounds, we learn noisy-OR Bayesian networks orders of magnitude faster than was possible with prior Monte Carlo learning algorithms, and provide a new tool for understanding large-scale binary data.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
