Interacting Langevin Diffusions: Gradient Structure And Ensemble Kalman Sampler

Garbuno-Iñigo, Alfredo; Hoffmann, Franca; Li, Wuchen; Stuart, Andrew M.

doi:10.48550/arxiv.1903.08866

Cited by 8 publications

(37 citation statements)

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“…We refer to this class of methods for calibration, collectively, as ensemble Kalman inversion (EKI) methods and note that pseudo-code for a variety of the methods may be found in [2]. An approach to using ensemble-based methods to produce approximate samples from the Bayesian posterior distribution on the unknown parameters is described in [22]; we refer to this method as ensemble Kalman sampling (EKS). Either of these approaches, EKI or EKS, may be used in the calibration step of our approximate Bayesian inversion method.…”

Section: Literature Reviewmentioning

confidence: 99%

Calibrate, Emulate, Sample

Cleary,

Garbuno-Inigo,

Lan

et al. 2020

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Many parameter estimation problems arising in applications are best cast in the framework of Bayesian inversion. This allows not only for an estimate of the parameters, but also for the quantification of uncertainties in the estimates. Often in such problems the parameterto-data map is very expensive to evaluate, and computing derivatives of the map, or derivative-adjoints, may not be feasible. Additionally, in many applications only noisy evaluations of the map may be available. We propose an approach to Bayesian inversion in such settings that builds on the derivative-free optimization capabilities of ensemble Kalman inversion methods. The overarching approach is to first use ensemble Kalman sampling (EKS) to calibrate the unknown parameters to fit the data; second, to use the output of the EKS to emulate the parameter-to-data map; third, to sample from an approximate Bayesian posterior distribution in which the parameter-to-data map is replaced by its emulator. This results in a principled approach to approximate Bayesian inference that requires only a small number of evaluations of the (possibly noisy approximation of the) parameter-to-data map. It does not require derivatives of this map, but instead leverages the documented power of ensemble Kalman methods. Furthermore, the EKS has the desirable property that it evolves the parameter ensembles towards the regions in which the bulk of the parameter posterior mass is located, thereby locating them well for the emulation phase of the methodology. In essence, the EKS methodology provides a cheap solution to the design problem of where to place points in parameter space to efficiently train an emulator of the parameter-to-data map for the purposes of Bayesian inversion.

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Section: Literature Reviewmentioning

confidence: 99%

Calibrate, Emulate, Sample

Cleary,

Garbuno-Inigo,

Lan

et al. 2020

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“…In this regard, the prime example (and also historically the first one where these concepts were layed out, see [37]) is given by the overdamped Langevin dynamics [64,Section 4.5], the associated Fokker-Planck equation of which takes the form of a gradient flow evolution driven by the KL-divergence in the geometry induced by the quadratic Wasserstein distance. Recently, similar ideas have been pursued, replacing either the driving functional or the underlying geometry, see, for instance, [3,23,29,30,44,67,77].…”

Section: Introductionmentioning

confidence: 99%

Stein Variational Gradient Descent: many-particle and long-time asymptotics

Nüsken¹,

Renger²

2021

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Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of Γ-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

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“…Hence, many MCMC methods can be viewed as Wasserstein gradient descent methods. In recent years, there are also several generalized Wasserstein metrics, such as Stein metric (Liu and Wang, 2016;Liu, 2017), Hessian transport (mobility) metrics (Carrillo et al, 2010;Dolbeault et al, 2009;Li and Ying, 2019) and Kalman-Wasserstein metric (Garbuno-Inigo et al, 2019). These metrics introduce various first-order methods with sampling efficient properties.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, the Stein variational gradient descent (Liu and Wang, 2016, SVGD) introduces a kernelized interacting Langevin dynamics. The Kalman-Wasserstein metric introduces a particular mean-field interacting Langevin dynamics (Garbuno-Inigo et al, 2019), known as ensemble Kalman sampling. On the other hand, many approaches design fast algorithms on modified Langevin dynamics.…”

Section: Introductionmentioning

confidence: 99%

Information Newton's flow: second-order optimization method in probability space

Wang,

2020

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We introduce a framework for Newton's flows in probability space with information metrics, named information Newton's flows. Here two information metrics are considered, including both the Fisher-Rao metric and the Wasserstein-2 metric. Several examples of information Newton's flows for learning objective/loss functions are provided, such as Kullback-Leibler (KL) divergence, Maximum mean discrepancy (MMD), and cross entropy. The asymptotic convergence results of proposed Newton's methods are provided. A known fact is that overdamped Langevin dynamics correspond to Wasserstein gradient flows of KL divergence. Extending this fact to Wasserstein Newton's flows of KL divergence, we derive Newton's Langevin dynamics. We provide examples of Newton's Langevin dynamics in both one-dimensional space and Gaussian families. For the numerical implementation, we design sampling efficient variational methods to approximate Wasserstein Newton's directions. Several numerical examples in Gaussian families and Bayesian logistic regression are shown to demonstrate the effectiveness of the proposed method.

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Interacting Langevin Diffusions: Gradient Structure And Ensemble Kalman Sampler

Cited by 8 publications

References 0 publications

Calibrate, Emulate, Sample

Calibrate, Emulate, Sample

Stein Variational Gradient Descent: many-particle and long-time asymptotics

Information Newton's flow: second-order optimization method in probability space

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