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Title: Non-asymptotic bounds for modified tamed unadjusted Langevin algorithm in non-convex setting
Authors: Ng, Matthew Cheng En
Keywords: Science::Mathematics::Probability theory
Science::Mathematics::Applied mathematics
Issue Date: 2022
Publisher: Nanyang Technological University
Source: Ng, M. C. E. (2022). Non-asymptotic bounds for modified tamed unadjusted Langevin algorithm in non-convex setting. Final Year Project (FYP), Nanyang Technological University, Singapore.
Abstract: We consider the problem of sampling from a target distribution $\pi_\beta$ on $\mathbb{R}^d$ with density proportional to $\theta\mapsto e^{-\beta U(\theta)}$ 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 recent literature to sample from target distributions with the gradient of the potential $U$ being super-linear. 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 non-convex setting, 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 $\pi_\beta$. We obtain resepctive rates of convergence $\mathcal{O}(\lambda)$ and $\mathcal{O}(\lambda^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $\lambda$. Numerical simulations which support our theoretical findings are presented.
Schools: School of Physical and Mathematical Sciences 
Fulltext Permission: restricted
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Student Reports (FYP/IA/PA/PI)

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