Implicit Langevin Algorithms for Sampling From Log-concave Densities

Hodgkinson, Liam; Salomone, Robert; Roosta, Fred

doi:10.48550/arxiv.1903.12322

Cited by 4 publications

(7 citation statements)

References 19 publications

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“…Also moving to integrators that are contractive for larger step sizes improves the performance for large condition numbers. This is consistent with what has been suggested in the literature [19,27].…”

Section: Introductionsupporting

confidence: 94%

See 1 more Smart Citation

Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

Sanz‐Serna¹,

Zygalakis²

2021

Preprint

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We present a framework that allows for the non-asymptotic study of the 2-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyse a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a d-dimensional strongly log-concave distribution with condition number κ, the algorithm is shown to produce with an O κ 5/4 d 1/4 ǫ −1/2 complexity samples from a distribution that, in Wasserstein distance, is at most ǫ > 0 away from the target distribution.

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Section: Introductionsupporting

confidence: 94%

“…These results have been extended to the Wasserstein distance W 2 in e.g. [14,15,16,17,18], while the paper [19] obtains similar bounds for implicit methods applied to (1.1). Similar nonasymptotic analyses for the case of the underdamped Langevin equation appear in [20,21,22,23,24].…”

Section: Introductionmentioning

confidence: 71%

Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

Sanz‐Serna¹,

Zygalakis²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Adaptive solvers prove particularly effective, although, as pointed out in Gholami et al (2019), the backward solve (4) can often run into stability issues, suggesting a Rosenbrock or other implicit approach (Hairer & Wanner, 1996). We point out that the same is also true in the stochastic setting; see Hodgkinson et al (2019), for example. For further implementation details concerning continuous normalizing flows, we refer to Grathwohl et al (2018).…”

Section: ∇L(z(t ))mentioning

confidence: 69%

Stochastic Normalizing Flows

Hodgkinson,

van der Heide,

Roosta

et al. 2020

Preprint

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We introduce stochastic normalizing flows, an extension of continuous normalizing flows for maximum likelihood estimation and variational inference (VI) using stochastic differential equations (SDEs). Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabling efficient training of neural SDEs as random neural ordinary differential equations. These SDEs can be used for constructing efficient Markov chains to sample from the underlying distribution of a given dataset. Furthermore, by considering families of targeted SDEs with prescribed stationary distribution, we can apply VI to the optimization of hyperparameters in stochastic MCMC.

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“…Proof. Our proof resembles that of [30,Proposition 1]. Let P denote the transition operator of {X k } ∞ k=1 , and let V s (x) = e s x .…”

Section: The Stein Correctionmentioning

confidence: 90%

“…. , w n ) that minimize the KSD (30), that is, the solution to the constrained quadratic program arg min w w K π w :…”

Section: Stein Importance Samplingmentioning

confidence: 99%

The reproducing Stein kernel approach for post-hoc corrected sampling

Hodgkinson,

Salomone,

Roosta

2020

Preprint

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Stein importance sampling [42] is a widely applicable technique based on kernelized Stein discrepancy [43], which corrects the output of approximate sampling algorithms by reweighting the empirical distribution of the samples. A general analysis of this technique is conducted for the previously unconsidered setting where samples are obtained via the simulation of a Markov chain, and applies to an arbitrary underlying Polish space. We prove that Stein importance sampling yields consistent estimators for quantities related to a target distribution of interest by using samples obtained from a geometrically ergodic Markov chain with a possibly unknown invariant measure that differs from the desired target. The approach is shown to be valid under conditions that are satisfied for a large number of unadjusted samplers, and is capable of retaining consistency when data subsampling is used. Along the way, a universal theory of reproducing Stein kernels is established, which enables the construction of kernelized Stein discrepancy on general Polish spaces, and provides sufficient conditions for kernels to be convergence-determining on such spaces. These results are of independent interest for the development of future methodology based on kernelized Stein discrepancies.

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Implicit Langevin Algorithms for Sampling From Log-concave Densities

Cited by 4 publications

References 19 publications

Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

Stochastic Normalizing Flows

The reproducing Stein kernel approach for post-hoc corrected sampling

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