On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions.
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- On the Convergence of the Metropolis Algorithm with Fixed-Order
Final published version, 3.07 MB, PDF document
The Metropolis algorithm is arguably the most fundamental Markov chain Monte Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the desired distribution in the case of multivariate binary distributions (e.g., Ising models or stochastic neural networks such as Boltzmann machines) if the variables (sites or neurons) are updated in a fixed order, a setting commonly used in practice. The reason is that the corresponding Markov chain may not be irreducible. We propose a modified Metropolis transition operator that behaves almost always identically to the standard Metropolis operator and prove that it ensures irreducibility and convergence to the limiting distribution in the multivariate binary case with fixed-order updates. The result provides an explanation for the behaviour of Metropolis MCMC in that setting and closes a long-standing theoretical gap. We experimentally studied the standard and modified Metropolis operator for models where they actually behave differently. If the standard algorithm also converges, the modified operator exhibits similar (if not better) performance in terms of convergence speed.
Original language | English |
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Title of host publication | Proceedings of The 24th International Conference on Artificial Intelligence and Statistic |
Publisher | PMLR |
Publication date | 2021 |
Pages | 469-477 |
Publication status | Published - 2021 |
Event | 24th International Conference on Artificial Intelligence and Statistics (AISTATS 2021) - San Diego, United States Duration: 13 Apr 2021 → 15 Apr 2021 |
Conference
Conference | 24th International Conference on Artificial Intelligence and Statistics (AISTATS 2021) |
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Land | United States |
By | San Diego |
Periode | 13/04/2021 → 15/04/2021 |
Series | Proceedings of Machine Learning Research |
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Volume | 130 |
ISSN | 1938-7228 |
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