SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence

Chewi, Sinho; Gouic, Thibaut Le; Chen, Lu; Maunu, Tyler; Rigollet, Philippe

doi:10.48550/arxiv.2006.02509

Cited by 4 publications

(8 citation statements)

References 16 publications

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“…In the above display, F stands for an appropriate free energy functional, |•| ρt and |•| ρ * t for suitable dual norms, and ε > 0 is assumed to be small. Moreover, Onsager and Machlup demonstrated that these constituents define a corresponding gradient flow structure 13 . Note that the exponent in (53) has the dimensions of a free energy (ignoring the Boltzmann constant and the constant temperature), which is consistent with the Boltzmann-Gibbs-Helmholtz free energy as described above.…”

Section: Connecting Gradient Flows To Large Deviationsmentioning

confidence: 99%

“…As stated in the Introduction, any evolution equation of gradient flow type in fact admits many other non-equivalent gradient flow structures [20]. In the case of the Stein PDE (8) this phenomenon is exemplified by the structures proposed in [44,23] and [13]. However, each gradient flow structure is related to a particular form of the noise in the corresponding interacting particle system.…”

Section: Connecting Gradient Flows To Large Deviationsmentioning

confidence: 99%

“…The gradient flow structure referred to above is not uniquely determined by (8); in fact the existence of one particular gradient flow structure implies that the PDE (8) admits infinitely many gradient flow structures [20]. For a particular example see [13], replacing the KL-by the χ 2 -divergence. Our first main result shows that the KL-gradient flow structure is naturally connected to the noise contribution in (6), bridging between the MCMC and VI viewpoints:…”

Section: A Connection Between Mcmc and VI Rooted In Statistical Physicsmentioning

confidence: 99%

“…Based on this, the authors of [39] developed nonasymptotic bounds in discrete time as well as propagation of chaos results (the latter of which unfortunately are not uniform in time). We would also like to mention the work [49] that rigorously establishes well-posednedness as well as convergence of the Stein PDE (8), and the work [13] that establishes an alternative gradient flow structure to the one considered in this paper.…”

Section: Previous Workmentioning

confidence: 99%

“…In what follows, we formally equip the set M defined in (13) with the structure of a Riemannian manifold, following [23,Section 4], where the reader is referred to for further details. To start with, recall the operators T k,ρ from (16), that can be extended to self-adjoint, nonnegative definite, and compact operators on L 2 (ρ), see [76,Section 4.3].…”

Section: Formal Riemannian Structure and Associated Gradientmentioning

confidence: 99%

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

Nüsken¹,

Renger²

2021

Preprint

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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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Section: Connecting Gradient Flows To Large Deviationsmentioning

confidence: 99%

Section: Connecting Gradient Flows To Large Deviationsmentioning

confidence: 99%

Section: A Connection Between Mcmc and VI Rooted In Statistical Physicsmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Section: Formal Riemannian Structure and Associated Gradientmentioning

confidence: 99%

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

Nüsken¹,

Renger²

2021

Preprint

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Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization

Yang,

Zhang,

Chen

et al. 2020

Preprint

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We consider the optimization problem of minimizing a functional defined over a family of probability distributions, where the objective functional is assumed to possess a variational form. Such a distributional optimization problem arises widely in machine learning and statistics, with Monte-Carlo sampling, variational inference, policy optimization, and generative adversarial network as examples. For this problem, we propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent over the manifold of probability distributions via iteratively pushing a set of particles. Specifically, we prove that moving along the geodesic in the direction of functional gradient with respect to the second-order Wasserstein distance is equivalent to applying a pushforward mapping to a probability distribution, which can be approximated accurately by pushing a set of particles. Specifically, in each iteration of variational transport, we first solve the variational problem associated with the objective functional using the particles, whose solution yields the Wasserstein gradient direction. Then we update the current distribution by pushing each particle along the direction specified by such a solution. By characterizing both the statistical error incurred in estimating the Wasserstein gradient and the progress of the optimization algorithm, we prove that when the objective functional satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly to the global minimum of the objective functional up to a certain statistical error, which decays to zero sublinearly as the number of particles goes to infinity.

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On Stein Variational Neural Network Ensembles

D’Angelo¹,

Fortuin²,

Wenzel³

2021

Preprint

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Ensembles of deep neural networks have achieved great success recently, but they do not offer a proper Bayesian justification. Moreover, while they allow for averaging of predictions over several hypotheses, they do not provide any guarantees for their diversity, leading to redundant solutions in function space. In contrast, particle-based inference methods, such as Stein variational gradient descent (SVGD), offer a Bayesian framework, but rely on the choice of a kernel to measure the similarity between ensemble members. In this work, we study different SVGD methods operating in the weight space, function space, and in a hybrid setting. We compare the SVGD approaches to other ensembling-based methods in terms of their theoretical properties and assess their empirical performance on synthetic and real-world tasks. We find that SVGD using functional and hybrid kernels can overcome the limitations of deep ensembles. It improves on functional diversity and uncertainty estimation and approaches the true Bayesian posterior more closely. Moreover, we show that using stochastic SVGD updates, as opposed to the standard deterministic ones, can further improve the performance.Preprint. Under review.

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SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence

Cited by 4 publications

References 16 publications

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

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

Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization

On Stein Variational Neural Network Ensembles

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