Faster logconcave sampling from a cold start in high dimension
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912360485093376 |
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| author | Kook, Yunbum Vempala, Santosh S. |
| author_facet | Kook, Yunbum Vempala, Santosh S. |
| contents | We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies.
Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular $q$-Rényi divergence for $q=\widetilde{\mathcal{O}}(1)$, whereas previous analyses required stringent $\infty$-Rényi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lovász and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Faster logconcave sampling from a cold start in high dimension Kook, Yunbum Vempala, Santosh S. Data Structures and Algorithms Machine Learning Functional Analysis Statistics Theory We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular $q$-Rényi divergence for $q=\widetilde{\mathcal{O}}(1)$, whereas previous analyses required stringent $\infty$-Rényi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lovász and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix. |
| title | Faster logconcave sampling from a cold start in high dimension |
| topic | Data Structures and Algorithms Machine Learning Functional Analysis Statistics Theory |
| url | https://arxiv.org/abs/2505.01937 |