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Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials

Lorenz Fruehwirth, Andreas Habring

Abstract

This article is concerned with sampling from Gibbs distributions $π(x)\propto e^{-U(x)}$ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials $U(x)$ which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density $π$ in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate $π$ for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.

Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials

Abstract

This article is concerned with sampling from Gibbs distributions using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.

Paper Structure

This paper contains 24 sections, 16 theorems, 75 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization of the Article
  4. Related Works
  5. Langevin sampling from non-smooth potentials
  6. Potential smoothing and similar techniques
  7. Methods not employing Langevin dynamics
  8. Notation and Preliminaries
  9. Analysis of the Continuous Time Dynamics
  10. Existence and Uniqueness of SDE Solutions
  11. Ergodicity of the Continuous Time Process
  12. Identifying the stationary distribution
  13. Discretizations of the Langevin Dynamics
  14. Explicit Euler-Maruyama Scheme
  15. Semi-implicit Euler-Maruyama Scheme
  16. ...and 9 more sections

Key Result

Theorem 1

Let $U: \mathbb{R}^d \rightarrow \mathbb{R}$ satisfy Ass:Potentials. Then, for any initial value $X_0\sim \mu \in \mathcal{P}_2(\mathbb{R}^d)$, the SDE eq:sde has a unique strong solution $(X_t)_{t \geq 0}$. The solution $(X_t)_{t \geq 0}$ is a continuous semimartingale with $\mathbb{E} \left[ \sup_

Figures (6)

  • Figure 1: Potential $U(x)=F(x)+G(x)$. On the left we show the Wasserstein-$2$ and on the right the total variation distance between samples and target $\pi$ for different step sizes and methods.
  • Figure 2: Potential $U(x)=F(x)+G(Kx)$ with linear operator in $G$. On the left we show the Wasserstein-$2$ and on the right the total variation distance between samples and target $\pi$ for different step sizes and methods.
  • Figure 3: Denoising: estimated expected values and variances. From left to right: Corrupted image $y$, expected values computed with BP, MYULA, and the proposed explicit scheme. Then variances computed with BP, MYULA, and the proposed explicit scheme. For MYULA and the proposed method we compute the statistics using the following 5e5 iterates after a burnin phase of 5e5 iterations.
  • Figure 4: Denoising: $L_2$-error of estimated expected value (left) and variance (right) of the proposed explicit scheme and MYULA each compared to BP results for the peppers image. We use a burnin phase of 5e5. The symbols $\bar{x}_k,\sigma_k$ denote the emprical expected value and variance using $k$ successive iterates, $\bar{x},\sigma$ the estimates from BP.
  • Figure 5: Deconvolution: estimated expected values and variances. From left to right: Corrupted image $y$, expected values computed with BP, MYULA, and the proposed explicit scheme. Then variances computed with BP, MYULA, and the proposed explicit scheme. For MYULA and the proposed method we compute the statistics using the following 5e5 iterates after a burnin phase of 5e5 iterations.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Remark 4.2
  • Remark 4.3
  • Theorem 1
  • proof
  • Remark 4.4
  • Lemma 4.5
  • proof
  • Remark 4.6
  • Lemma 4.7
  • proof
  • ...and 31 more