Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions
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| Format: | Preprint |
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2025
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| _version_ | 1866911559157022720 |
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| author | Fornasier, Massimo Sun, Lukang Ward, Rachel |
| author_facet | Fornasier, Massimo Sun, Lukang Ward, Rachel |
| contents | In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions $\mathcal{L}$. We establish well-posedness and regularity of the law $ρ_t$ of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of $ρ_t$ under truly minimal assumptions on $\mathcal{L}$. In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is $\mathcal{O}(1/t)$. Under a Polyak-Lojasiewicz (PL) condition on $\mathcal{L}$, we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate $\mathcal{O}(1/t)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions Fornasier, Massimo Sun, Lukang Ward, Rachel Analysis of PDEs Probability In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions $\mathcal{L}$. We establish well-posedness and regularity of the law $ρ_t$ of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of $ρ_t$ under truly minimal assumptions on $\mathcal{L}$. In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is $\mathcal{O}(1/t)$. Under a Polyak-Lojasiewicz (PL) condition on $\mathcal{L}$, we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate $\mathcal{O}(1/t)$. |
| title | Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2510.24925 |