Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909222367657984 |
|---|---|
| author | Neufeld, Ariel En, Matthew Ng Cheng Zhang, Ying |
| author_facet | Neufeld, Ariel En, Matthew Ng Cheng Zhang, Ying |
| contents | We consider the problem of sampling from a high-dimensional target distribution $π_β$ on $\mathbb{R}^d$ with density proportional to $θ\mapsto e^{-βU(θ)}$ using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [Bro+19] to sample from such target distributions with the gradient of the potential $U$ being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential $U$. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential $U$ and super-linearly growing gradient of $U$, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution $π_β$. We obtain respective rates of convergence $\mathcal{O}(λ)$ and $\mathcal{O}(λ^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $λ$. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_02600 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting Neufeld, Ariel En, Matthew Ng Cheng Zhang, Ying Probability Numerical Analysis Optimization and Control Statistics Theory Computation We consider the problem of sampling from a high-dimensional target distribution $π_β$ on $\mathbb{R}^d$ with density proportional to $θ\mapsto e^{-βU(θ)}$ using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [Bro+19] to sample from such target distributions with the gradient of the potential $U$ being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential $U$. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential $U$ and super-linearly growing gradient of $U$, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution $π_β$. We obtain respective rates of convergence $\mathcal{O}(λ)$ and $\mathcal{O}(λ^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $λ$. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm. |
| title | Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting |
| topic | Probability Numerical Analysis Optimization and Control Statistics Theory Computation |
| url | https://arxiv.org/abs/2207.02600 |