Simple parallel estimation of the partition ratio for Gibbs distributions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912999601602560 |
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| author | Harris, David G. Kolmogorov, Vladimir |
| author_facet | Harris, David G. Kolmogorov, Vladimir |
| contents | We consider the problem of estimating the partition function $Z(β)=\sum_x \exp(β(H(x))$ of a Gibbs distribution with the Hamiltonian $H:Ω\rightarrow\{0\}\cup[1,n]$. As shown in [Harris & Kolmogorov 2024], the log-ratio $q=\ln (Z(β_{\max})/Z(β_{\min}))$ can be estimated with accuracy $ε$ using $O(\frac{q \log n}{ε^2})$ calls to an oracle that produces a sample from the Gibbs distribution for parameter $β\in[β_{\min},β_{\max}]$. That algorithm is inherently sequential, or {\em adaptive}: the queried values of $β$ depend on previous samples. Recently, [Liu, Yin & Zhang 2024] developed a non-adaptive version that needs $O( q (\log^2 n) (\log q + \log \log n + ε^{-2}) )$ samples.
We improve the number of samples to $O(\frac{q \log^2 n}{ε^2})$ for a non-adaptive algorithm, and to $O(\frac{q \log n}{ε^2})$ for an algorithm that uses just two rounds of adaptivity (matching the complexity of the sequential version). Furthermore, our algorithm simplifies previous techniques. In particular, we use just a single estimator, whereas methods in [Harris & Kolmogorov 2024, Liu, Yin & Zhang 2024] employ two different estimators for different regimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_18324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple parallel estimation of the partition ratio for Gibbs distributions Harris, David G. Kolmogorov, Vladimir Probability Data Structures and Algorithms We consider the problem of estimating the partition function $Z(β)=\sum_x \exp(β(H(x))$ of a Gibbs distribution with the Hamiltonian $H:Ω\rightarrow\{0\}\cup[1,n]$. As shown in [Harris & Kolmogorov 2024], the log-ratio $q=\ln (Z(β_{\max})/Z(β_{\min}))$ can be estimated with accuracy $ε$ using $O(\frac{q \log n}{ε^2})$ calls to an oracle that produces a sample from the Gibbs distribution for parameter $β\in[β_{\min},β_{\max}]$. That algorithm is inherently sequential, or {\em adaptive}: the queried values of $β$ depend on previous samples. Recently, [Liu, Yin & Zhang 2024] developed a non-adaptive version that needs $O( q (\log^2 n) (\log q + \log \log n + ε^{-2}) )$ samples. We improve the number of samples to $O(\frac{q \log^2 n}{ε^2})$ for a non-adaptive algorithm, and to $O(\frac{q \log n}{ε^2})$ for an algorithm that uses just two rounds of adaptivity (matching the complexity of the sequential version). Furthermore, our algorithm simplifies previous techniques. In particular, we use just a single estimator, whereas methods in [Harris & Kolmogorov 2024, Liu, Yin & Zhang 2024] employ two different estimators for different regimes. |
| title | Simple parallel estimation of the partition ratio for Gibbs distributions |
| topic | Probability Data Structures and Algorithms |
| url | https://arxiv.org/abs/2505.18324 |