On the geometry of Stein variational gradient descent

Duncan, Anthony; Nuesken, N.; Szpruch, Łukasz

doi:10.48550/arxiv.1912.00894

Cited by 25 publications

(60 citation statements)

References 78 publications

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“…Constructing numerical approximations for ( 6)-( 7) beyond the constant-in-space approximation is a topic of ongoing research. We mention in particular the approach developed in [97] and analysed in [85] based on diffusion maps as well as the method from [79] based on the Stein geometry [41,81].…”

Section: From the Filtering Problem To The Mckean-vlasov Equationmentioning

confidence: 99%

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Coghi¹,

Nilssen²,

Nüsken³

et al. 2021

Preprint

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Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.

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Section: From the Filtering Problem To The Mckean-vlasov Equationmentioning

confidence: 99%

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Coghi¹,

Nilssen²,

Nüsken³

et al. 2021

Preprint

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“…The third option we consider is the non-parametric updates, which we derive approximating the energy gradient in RKHS H with kernel k. To minimize the KL-divergence we first take the Frechet derivative w.r.t. the energy E along some direction h and use the fact that H is actually dense in L 2 q (Duncan et al, 2019). Then, using the reproducing property of k, we can formulate the directional derivative as an action of a linear operator:…”

Section: Algorithm 1 Methods α βmentioning

confidence: 99%

Particle Dynamics for Learning EBMs

Neklyudov¹,

Jaini²,

Welling³

2021

Preprint

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Energy-based modeling is a promising approach to unsupervised learning, which yields many downstream applications from a single model. The main difficulty in learning energy-based models with the "contrastive approaches" is the generation of samples from the current energy function at each iteration. Many advances have been made to accomplish this subroutine cheaply. Nevertheless, all such sampling paradigms run MCMC targeting the current model, which requires infinitely long chains to generate samples from the true energy distribution and is problematic in practice. This paper proposes an alternative approach to getting these samples and avoiding crude MCMC sampling from the current model. We accomplish this by viewing the evolution of the modeling distribution as (i) the evolution of the energy function, and (ii) the evolution of the samples from this distribution along some vector field. We subsequently derive this time-dependent vector field such that the particles following this field are approximately distributed as the current density model. Thereby we match the evolution of the particles with the evolution of the energy function prescribed by the learning procedure. Importantly, unlike Monte Carlo sampling, our method targets to match the current distribution in a finite time. Finally, we demonstrate its effectiveness empirically comparing to MCMC-based learning methods.

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“…Stein Variational Gradient Descent (SVGD) Liu and Wang [2016], Liu [2017] is an alternative to the Langevin algorithm and has been applied in several contexts in machine learning, including Reinforcement Learning , sequential decision making Zhang et al [2018, Generative Adversarial Networks Tao et al [2019], Variational Auto Encoders Pu et al [2017], and Federated Learning Kassab and Simeone [2020]. However, the theoretical understanding of SVGD is limited compared to that of Langevin algorithm Lu et al [2019], Duncan et al [2019], Liu [2017], Chewi et al [2020], Nüsken and Renger [2021]. In particular, the first complexity result of SVGD, due to Korba et al [2020, Corollary 6], appeared only recently, and relies on an assumption on the trajectory of the algorithm, which cannot be checked prior to running the algorithm.…”

Section: Stein Variational Gradient Descent (Svgd)mentioning

confidence: 99%

A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1

Salim¹,

Sun²,

Richtárik³

2021

Preprint

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We study the complexity of Stein Variational Gradient Descent (SVGD), which is an algorithm to sample from π(x) ∝ exp(−F (x)) where F smooth and nonconvex. We provide a clean complexity bound for SVGD in the population limit in terms of the Stein Fisher Information (or squared Kernelized Stein Discrepancy), as a function of the dimension of the problem d and the desired accuracy ε. Unlike existing work, we do not make any assumption on the trajectory of the algorithm. Instead, our key assumption is that the target distribution satisfies Talagrand's inequality T1.

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On the geometry of Stein variational gradient descent

Cited by 25 publications

References 78 publications

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Particle Dynamics for Learning EBMs

A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1

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