The ballistic limit of the log-Sobolev constant equals the Polyak-Łojasiewicz constant
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915024541319168 |
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| author | Chewi, Sinho Stromme, Austin J. |
| author_facet | Chewi, Sinho Stromme, Austin J. |
| contents | The Polyak-Lojasiewicz (PL) constant of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ characterizes the best exponential rate of convergence of gradient flow for $f$, uniformly over initializations. Meanwhile, in the theory of Markov diffusions, the log-Sobolev (LS) constant plays an analogous role, governing the exponential rate of convergence for the Langevin dynamics from arbitrary initialization in the Kullback-Leibler divergence. We establish a new connection between optimization and sampling by showing that the low temperature limit $\lim_{t\to 0^+} t^{-1} C_{\mathsf{LS}}(μ_t)$ of the LS constant of $μ_t \propto \exp(-f/t)$ is exactly the PL constant of $f$, under mild assumptions. In contrast, we show that the corresponding limit for the Poincaré constant is the inverse of the smallest eigenvalue of $\nabla^2 f$ at the minimizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ballistic limit of the log-Sobolev constant equals the Polyak-Łojasiewicz constant Chewi, Sinho Stromme, Austin J. Probability Functional Analysis Optimization and Control The Polyak-Lojasiewicz (PL) constant of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ characterizes the best exponential rate of convergence of gradient flow for $f$, uniformly over initializations. Meanwhile, in the theory of Markov diffusions, the log-Sobolev (LS) constant plays an analogous role, governing the exponential rate of convergence for the Langevin dynamics from arbitrary initialization in the Kullback-Leibler divergence. We establish a new connection between optimization and sampling by showing that the low temperature limit $\lim_{t\to 0^+} t^{-1} C_{\mathsf{LS}}(μ_t)$ of the LS constant of $μ_t \propto \exp(-f/t)$ is exactly the PL constant of $f$, under mild assumptions. In contrast, we show that the corresponding limit for the Poincaré constant is the inverse of the smallest eigenvalue of $\nabla^2 f$ at the minimizer. |
| title | The ballistic limit of the log-Sobolev constant equals the Polyak-Łojasiewicz constant |
| topic | Probability Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2411.11415 |