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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.13085 |
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| _version_ | 1866917085381132288 |
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| author | Shen, Tian Su, Zhonggen |
| author_facet | Shen, Tian Su, Zhonggen |
| contents | The task of sampling from a high-dimensional distribution $π$ on $\R^d$ is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on $\R^d$ \begin{align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{2}dB_t, \end{align*} under mild conditions, it admits $π(\dif x)\propto \exp(-U(x))\dif x$ as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution $π$. In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution $π_η$ ($η$ is the step size) and obtain the convergence rate of PRLMC to $π_η$ in total variation distance. Then we establish a sharp error bound between $π_η$ and $π$ under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to $π$ which is nearly optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13085 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-asymptotic Analysis of Poisson randomized midpoint Langevin Monte Carlo Shen, Tian Su, Zhonggen Statistics Theory Probability The task of sampling from a high-dimensional distribution $π$ on $\R^d$ is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on $\R^d$ \begin{align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{2}dB_t, \end{align*} under mild conditions, it admits $π(\dif x)\propto \exp(-U(x))\dif x$ as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution $π$. In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution $π_η$ ($η$ is the step size) and obtain the convergence rate of PRLMC to $π_η$ in total variation distance. Then we establish a sharp error bound between $π_η$ and $π$ under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to $π$ which is nearly optimal. |
| title | Non-asymptotic Analysis of Poisson randomized midpoint Langevin Monte Carlo |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2511.13085 |