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arXiv:1705.06724 · Physical review. B./Physical review. B

Self-learning Monte Carlo method: Continuous-time algorithm

Yuki Nagai, Huitao Shen, Yang Qi, Junwei Liu, Liang Fu

Published Oct 3, 2017• 61 citations
arXivHTMLPDF

Abstract

The self-learning Monte Carlo (SLMC) method speeds up the Monte Carlo simulation by designing and training an effective model to propose efficient global updates. We implement the continuous-time quantum Monte Carlo algorithm for quantum impurity models in the framework of SLMC. We introduce and train a diagram generating function (DGF) to model the probability distribution of field configurations in the continuous imaginary time at all orders of diagrammatic expansion. By using the DGF to propose global updates, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.

DOI
10.1103/physrevb.96.161102

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•Jan 1, 2017•0 citations
The self-learning Monte Carlo (SLMC) method speeds up the Monte Carlo simulation by designing and training an effective model to propose efficient global updates. We implement the continuous-time quantum Monte Carlo algorithm for quantum impurity models in the framework of SLMC. We introduce and train a diagram generating function (DGF) to model the probability distribution of field configurations in the continuous imaginary time at all orders of diagrammatic expansion. By using the DGF to propose global updates, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.
The Self-Learning Monte Carlo (SLMC) method is a Monte Carlo approach that has emerged in recent years by integrating concepts from machine learning with conventional Monte Carlo techniques. Designed to accelerate the numerical study of interacting many-body systems, SLMC significantly improves sampling efficiency by constructing an effective model -- via machine learning methods -- based on configurations generated by conventional Monte Carlo methods and then proposes global updates based on the effective model. This enhancement leads to a substantial reduction in autocorrelation time, especially near the critical region, where traditional methods typically suffer from critical slowing down and increased computation complexity. Moreover, SLMC maintains statistical accuracy by implementing a cumulative update scheme that rigorously satisfies the detailed balance condition. And more recent applications have extended the SLMC to convolutional neural networks with applications not only in condensed matter physics but also high-energy physics, quantum chemistry, and quantum simulations. The generic applicability and high computational efficiency make SLMC a powerful and scalable framework for quantum Monte Carlo simulations of strongly correlated electron systems, extending the reach of numerical investigations beyond the limitations of conventional techniques.
The self-learning Monte Carlo (SLMC) method is a general algorithm to speedup MC simulations. Its efficiency has been demonstrated in various systems by introducing an effective model to propose global moves in the configuration space. In this paper, we show that deep neural networks can be naturally incorporated into SLMC, and without any prior knowledge can learn the original model accurately and efficiently. Demonstrated in quantum impurity models, we reduce the complexity for a local update from $O({\ensuremath{\beta}}^{2})$ in Hirsch-Fye algorithm to $O(\ensuremath{\beta}ln\ensuremath{\beta})$, which is a significant speedup especially for systems at low temperatures.
Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10–20 times speedup.
Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10--20 times speedup.
We develop the self-learning Monte Carlo (SLMC) method, a general-purpose numerical method recently introduced to simulate many-body systems, for studying interacting fermion systems. Our method uses a highly efficient update algorithm, which we design and dub ``cumulative update'', to generate new candidate configurations in the Markov chain based on a self-learned bosonic effective model. From a general analysis and a numerical study of the double exchange model as an example, we find that the SLMC with cumulative update drastically reduces the computational cost of the simulation, while remaining statistically exact. Remarkably, its computational complexity is far less than the conventional algorithm with local updates.
As a new general-purpose Monte Carlo method, self-learning Monte Carlo (SLMC) has been numerically demonstrated in boson systems. In this work, we propose a new type of update algorithm, dubbed cumulative update, which could be naturally integrated into SLMC. Based on many local updates in the self-learned effective model, cumulative update efficiently proposes a global move with a high acceptance probability in the original model. Cumulative update can reduce the conventional computational cost of a full sweep O(N4) to O(N3), and also effectively reduce the auto-correlation time to be 1. We numerically show its efficiency through the well-known double-exchange model. With cumulative update, the SLMC could be several hundreds of times faster than the conventional local update method. By employing the cumulative update, SLMC can be generally used in any sign-problem-free Fermion systems and maximize the performance of Monte Carlo simulation.