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1. Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo

   Gürbüzbalaban, M. ; Gao, X. ; Hu, Y. ; Zhu, L. ( January 2021 , Journal of machine learning research)

   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 

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2. Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning

   Lamperski, A. ( August 2021 , Proceedings of Thirty Fourth Conference on Learning Theory)

   Belkin, M. ; Kpotufe, S. (Ed.)

   Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of 𝑂(𝑇−1/4(𝑙𝑜𝑔𝑇)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves 𝜖-suboptimal solutions, on average, provided that it is run for a time that is polynomial in 𝜖 and slightly super-exponential in the problem dimension. 

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3. Simulation‐based estimators of analytically intractable causal effects

   https://doi.org/10.1111/biom.13499  

   Linero, Antonio R. ( June 2021 , Biometrics)

   In causal inference problems, one is often tasked with estimating causal effects which are analytically intractable functionals of the data‐generating mechanism. Relevant settings include estimating intention‐to‐treat effects in longitudinal problems with missing data or computing direct and indirect effects in mediation analysis. One approach to computing these effects is to use theg‐formula implemented via Monte Carlo integration; when simulation‐based methods such as the nonparametric bootstrap or Markov chain Monte Carlo are used for inference, Monte Carlo integration must be nested within an already computationally intensive algorithm. We develop a widely‐applicable approach to accelerating this Monte Carlo integration step which greatly reduces the computational burden of existingg‐computation algorithms. We refer to our method as acceleratedg‐computation (AGC). The algorithms we present are similar in spirit to multiple imputation, but require removing within‐imputation variance from the standard error rather than adding it. We illustrate the use of AGC on a mediation analysis problem using a beta regression model and in a longitudinal clinical trial subject to nonignorable missingness using a Bayesian additive regression trees model.

    

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4. Efficient derivative-free Bayesian inference for large-scale inverse problems

   https://doi.org/10.1088/1361-6420/ac99fa  

   Huang, Daniel Zhengyu ; Huang, Jiaoyang ; Reich, Sebastian ; Stuart, Andrew M ( October 2022 , Inverse Problems)

   Abstract We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O ( 1 0 4 ) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator. Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model. Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically. The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems. 

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5. A fast asynchronous Markov chain Monte Carlo sampler for sparse Bayesian inference

   https://doi.org/10.1093/jrsssb/qkad078  

   Atchadé, Yves ; Wang, Liwei ( September 2023 , Journal of the Royal Statistical Society Series B: Statistical Methodology)

   We propose a very fast approximate Markov chain Monte Carlo sampling framework that is applicable to a large class of sparse Bayesian inference problems. The computational cost per iteration in several regression models is of order O(n(s+J)), where n is the sample size, s is the underlying sparsity of the model, and J is the size of a randomly selected subset of regressors. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin dynamics are employed. The algorithm is an extension of the asynchronous Gibbs sampler of Johnson et al. [(2013). Analyzing Hogwild parallel Gaussian Gibbs sampling. In Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS’13) (Vol. 2, pp. 2715–2723)], but can be viewed from a statistical perspective as a form of Bayesian iterated sure independent screening [Fan, J., Samworth, R., & Wu, Y. (2009). Ultrahigh dimensional feature selection: Beyond the linear model. Journal of Machine Learning Research, 10, 2013–2038]. We show that in high-dimensional linear regression problems, the Markov chain generated by the proposed algorithm admits an invariant distribution that recovers correctly the main signal with high probability under some statistical assumptions. Furthermore, we show that its mixing time is at most linear in the number of regressors. We illustrate the algorithm with several models.

    

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