Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913979385774080 |
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| author | Neufeld, Ariel Zhang, Ying |
| author_facet | Neufeld, Ariel Zhang, Ying |
| contents | In this paper, we propose two new algorithms, namely, aHOLA and aHOLLA, to sample from high-dimensional target distributions with possibly super-linearly growing potentials. We establish non-asymptotic convergence bounds for aHOLA in Wasserstein-1 and Wasserstein-2 distances with rates of convergence equal to $1+q/2$ and $1/2+q/4$, respectively, under a local Hölder condition with exponent $q\in(0,1]$ and a convexity at infinity condition on the potential of the target distribution. Similar results are obtained for aHOLLA under certain global continuity conditions and a dissipativity condition. Crucially, we achieve state-of-the-art rates of convergence of the proposed algorithms in the non-convex setting which are higher than those of the existing algorithms. Examples from high-dimensional sampling and logistic regression are presented, and numerical results support our main findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms Neufeld, Ariel Zhang, Ying Statistics Theory Probability Computation Machine Learning In this paper, we propose two new algorithms, namely, aHOLA and aHOLLA, to sample from high-dimensional target distributions with possibly super-linearly growing potentials. We establish non-asymptotic convergence bounds for aHOLA in Wasserstein-1 and Wasserstein-2 distances with rates of convergence equal to $1+q/2$ and $1/2+q/4$, respectively, under a local Hölder condition with exponent $q\in(0,1]$ and a convexity at infinity condition on the potential of the target distribution. Similar results are obtained for aHOLLA under certain global continuity conditions and a dissipativity condition. Crucially, we achieve state-of-the-art rates of convergence of the proposed algorithms in the non-convex setting which are higher than those of the existing algorithms. Examples from high-dimensional sampling and logistic regression are presented, and numerical results support our main findings. |
| title | Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms |
| topic | Statistics Theory Probability Computation Machine Learning |
| url | https://arxiv.org/abs/2405.05679 |