Mean Field Initialization of the Annealed Importance Sampling Algorithm for an Efficient Evaluation of the Partition Function Using Restricted Boltzmann Machines
Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this p...
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| Vydané v: | Entropy (Basel, Switzerland) Ročník 27; číslo 2; s. 171 |
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06.02.2025
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| Abstract | Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset. |
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| AbstractList | Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset. Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset.Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset. Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function . Obtaining the exact value of , though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the partition function of magnetic spin and spin-like systems mapped into RBMs using AIS. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of following an annealing process. We show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost. The procedures presented are not linked to any learning process, and therefore do not require a priori knowledge of a training dataset. |
| Audience | Academic |
| Author | Martí, Jordi Mazzanti, Ferran Prat Pou, Arnau Romero, Enrique |
| AuthorAffiliation | 2 Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona Tech, Campus Nord B4-B5, E-08034 Barcelona, Spain; eromero@cs.upc.edu 1 Departament de Física, Universitat Politècnica de Catalunya, Barcelona Tech, Campus Nord B4-B5, E-08034 Barcelona, Spain; arnau.prat.pou@upc.edu (A.P.P.); jordi.marti@upc.edu (J.M.) |
| AuthorAffiliation_xml | – name: 1 Departament de Física, Universitat Politècnica de Catalunya, Barcelona Tech, Campus Nord B4-B5, E-08034 Barcelona, Spain; arnau.prat.pou@upc.edu (A.P.P.); jordi.marti@upc.edu (J.M.) – name: 2 Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Barcelona Tech, Campus Nord B4-B5, E-08034 Barcelona, Spain; eromero@cs.upc.edu |
| Author_xml | – sequence: 1 givenname: Arnau orcidid: 0000-0002-1770-9636 surname: Prat Pou fullname: Prat Pou, Arnau – sequence: 2 givenname: Enrique surname: Romero fullname: Romero, Enrique – sequence: 3 givenname: Jordi orcidid: 0000-0002-3721-9634 surname: Martí fullname: Martí, Jordi – sequence: 4 givenname: Ferran surname: Mazzanti fullname: Mazzanti, Ferran |
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| Cites_doi | 10.1016/j.neunet.2012.06.003 10.1103/RevModPhys.67.279 10.1119/1.1707017 10.1126/science.1127647 10.1145/1273496.1273596 10.1017/CBO9780511623257 10.1162/neco_a_01420 10.1023/A:1008923215028 10.1063/1.4907883 10.1007/BF02980577 10.1103/PhysRevLett.96.120201 10.7551/mitpress/5236.001.0001 10.1016/j.artint.2019.103195 10.1016/j.jcp.2011.12.008 10.1103/PhysRev.65.117 10.1103/PhysRevLett.110.210603 10.1016/j.cpc.2010.10.031 10.1142/9789813232105_0006 10.1103/PhysRevE.103.013302 10.1016/0021-9991(76)90078-4 10.1561/2200000006 10.1103/PhysRevB.100.064304 10.1063/1.481926 10.1093/oso/9780198513940.001.0001 10.1103/PhysRevE.106.024127 10.1119/1.18168 10.1007/978-3-662-05052-1_5 10.21468/SciPostPhys.16.4.095 10.1073/pnas.1505664112 10.1103/PhysRevE.81.016707 10.1109/TPAMI.1984.4767596 |
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| Snippet | Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function... Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function .... |
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| StartPage | 171 |
| SubjectTerms | Algorithms annealed importance sampling Annealing Computing costs Energy Importance sampling magnetic systems Monte Carlo simulation partition function Partitions (mathematics) Probabilistic models Probability Probability distribution Restricted Boltzmann Machines Statistical analysis |
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| Title | Mean Field Initialization of the Annealed Importance Sampling Algorithm for an Efficient Evaluation of the Partition Function Using Restricted Boltzmann Machines |
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