Exponential convergence of Langevin distributions and their discrete approximations

By Gareth O. Roberts, Richard L. Tweedie. 1996

Subfields: Probabilistic & Bayesian Methods

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Citations in the corpus

Improved Techniques for Training Score-Based Generative Models (2020, methodology): “When α is sufficiently small and T is sufficiently large, the distribution of xT will be close to p(x) under some regularity conditions [6, 7].”
Plug and Play Language Models: A Simple Approach to Controlled Text Generation (2019, methodology): “We sample from the resulting combined model by following gradients in the latent representation space in a manner inspired by the approximate Metropolis-adjusted Langevin (MALA) (Roberts et al., 1996; Roberts & Rosenthal, 1998) sampler deployed in Nguyen et al. (2017).”
Handbook of Markov Chain Monte Carlo (2011, neutral): “See Roberts and Tweedie (1996) for a discussion of this in the context of the Langevin method (see Section 5.2).”
Adversarial score matching and improved sampling for image generation (2020, methodology): “…Ermon (2019) recently proposed a novel method of generating samples from a target distribution through a combination of Denoising Score Matching (DSM) (Hyvärinen, 2005; Vincent, 2011; Raphan and Simoncelli, 2011) and Annealed Langevin Sampling (ALS) (Welling and Teh, 2011; Roberts et al., 1996).”
A Tutorial on Thompson Sampling (2017, background): “This process has g(φ) as its unique stationary distribution, and under reasonable technical conditions, the distribution of φt converges rapidly to this stationary distribution [19].”
Measuring Sample Quality with Stein's Method (2015, methodology): “To corroborate our result, we use a Metropolis-adjusted Langevin chain [26] of length 107 as a surrogate Q⇤ for the target and compute several error measures for each sample Q: normalized probability error max”
Coresets for Scalable Bayesian Logistic Regression (2016, methodology): “For the synthetic (resp. real-world) data, adaptive MALA was run once on the full dataset for 100,000 (resp.”
Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models (2020, methodology): “…the Brownian motion term with the gradient of the log marginal density without modifying its Fokker-Planck equation (see Hoffman & Ma (2019) or the appendix of Wang & Li (2019)) and (2) the ﬁrst-order Langevin dynamic is known to be convergent to its stationary distribution (Roberts et al., 1996).”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, methodology): “Whileintheprobability community Langevin diffusions and their equilibrium distributions had also been around for some time (Kent 1978), it was the Roberts and Tweedie (1996a) paper (motivated by Besag 1994 comment on Grenander and Miller 1994) that brought the approach to the centre of interest of…”
Towards Automatic Model Comparison: An Adaptive Sequential Monte Carlo Approach (2013, methodology): “One possible approach is using the Metropolis adjusted Langevin algorithm (MALA; see Roberts and Tweedie (1996)).”
Measuring Sample Quality with Diffusions (2016, methodology): “To provide external support for the Stein discrepancy quantification, we generate a Metropolis-adjusted Langevin chain [77] of length 108 as a surrogate Q∗ for the target P and display several measures of expectation error between X ∼ Q and Z ∼ Q∗ in Figure 3: the normalized predictive error maxl |E”
Langevin Diffusions and Metropolis-Hastings Algorithms (2002, methodology): “As noted in Roberts and Tweedie (1996b), by the positive de®niteness condition lim inf jxj ?”
A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics (2017, background): “Another line of work in Bayesian statistics studies the Langevin Monte Carlo (LMC) method [28] which employs an alternative choice of noise term.”
Geometric ergodicity of the Bouncy Particle Sampler (2018, background): “As a consequence, no Metropolis ﬁlter is necessary to preserve the invariance of π , see [43, 13, 37, 41] and the reference therein.”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, methodology): “Asymptotic guarantees for overdamped Langevin MCMC was established much earlier in [8, 9].”
A Gentle Stochastic Thermostat for Molecular Dynamics (2009, result): “We state a basic result which relates ergodicity to regularity of solutions of the Fokker-Planck equation; for similar results see [18, 19, 8], as well as earlier works on ergodicity of Markov chains [21, 28, 27 ].”
Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients (2009, unknown): “Next we relate the divergence result (1.8) regarding ﬁnite time intervals to a divergence result of Mattingly, Stuart & Higham (2002) regarding inﬁnite time intervals (see also Theorem 3.2 in Roberts & Tweedie 1996 and Lemma 4.1 in Higham, Mao & Stuart 2003).”
Posterior Computation with the Gibbs Zig-Zag Sampler (2020, methodology): “This includes MH samplers that obtain efﬁcient joint proposals using gradient information, ranging from Hamiltonian Monte Carlo (HMC, Duane et al., 1987) to Metropolis-adjusted Langevin algorithms (Roberts and Tweedie, 1996).”