High-dimensional Bayesian inference via the unadjusted Langevin algorithm

By Alain Oliviero Durmus, Eric Moulines. Bernoulli, 2016

417 citations recorded by the source corpus.

We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density with respect to the Lebesgue measure on $\mathbb{R}^d$, known up to a normalization constant $x \mapsto \pi(x)= \mathrm{e}^{-U(x)}/\int_{\mathbb{R}^d} \mathrm{e}^{-U(y)} \mathrm{d} y$. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipschitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order $2$ and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e242caa5e60776ab2df1ed758df66b4020ae4ded

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Citations in the corpus

Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula (2018, methodology): “s to Appendix A. Remark 2.4. In the literature on multivariate probability approximations by µ, it is often assumed that µ is strongly log-concave. See, for example, Eq. (1) in [10], Assumption H2 in [13] and Theorem 2.1 in [24]. This corresponds to condition (2.3) with θ0 > 0 and θ1 = 0. We use ”
User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient (2017, methodology): “Note that if the initial value ϑ 0 = θ 0 is deterministic then, according to (Durmus and Moulines, 2016, Theorem 1), we have Finally, let us remark that if we choose h and K so that then we have W 2 ( ν K , π ) ≤ ε .”
Log-concave sampling: Metropolis-Hastings algorithms are fast! (2018, background): “However, in a series of recent works [11, 12, 7], it has been established that with a careful choice of step-size h and iteration count K, running the chain (6) for exactly K steps yields an iterate xK whose distribution is close to Π.”
Quantitative Central Limit Theorems for Discrete Stochastic Processes (2019, background): “Authors such as Dalalyan (2017) and Durmus and Moulines (2016) were able to prove quantitative convergence results for Langevin MCMC by bounding its discretization error from the Langevin SDE.”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “Its curl matrix Q = 0 makes its fGH flow comprise purely the gradient flow, allowing a rich study on its convergence (e.g. Durmus & Moulines (2016); Cheng & Bartlett (2017)).”
A survey of uncertainty in deep neural networks (2021, background): “Lastly, other recent trends are techniques that reduce the variance [160], [182] and bias [183], [184] arising from stochastic gradients.”