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International Symposium on Lattice Field Theory (LATTICE)

Make life simple: unleash the full power of the parallel tempering algorithm

Elmar Bittner, Andreas Nußbaumer, Wolfhard Janke E

Published Jun 23, 2010• 0 citations
Publisher

Abstract

We introduce a new update scheme to systematically improve the efficiency of parallel tempering Monte Carlo simulations by taking into account the temperature dependence of autocorrelation times.In contrast to previous attempts the temperatures are not dynamically adjusted but chosen in such a way that the acceptance rate for proposed exchanges of all adjacent replica is about 50%.We show that by adapting the number of sweeps between the parallel tempering moves to the canonical autocorrelation time, the average round-trip time of a replica between the lowest and the highest temperatures is significantly decreased and, therefore, the efficiency of the parallel tempering algorithm is considerably improved.We illustrate the new algorithm with results for the two-dimensional Ising model and propose a toy model to find the optimal parameter set for the parallel tempering routine within no time.

DOI
10.22323/1.091.0027

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We introduce a new update scheme to systematically improve the efficiency of parallel tempering Monte Carlo simulations by taking into account the temperature dependence of autocorrelation times.In contrast to previous attempts the temperatures are not dynamically adjusted but chosen in such a way that the acceptance rate for proposed exchanges of all adjacent replica is about 50%.We show that by adapting the number of sweeps between the parallel tempering moves to the canonical autocorrelation time, the average round-trip time of a replica between the lowest and the highest temperatures is significantly decreased and, therefore, the efficiency of the parallel tempering algorithm is considerably improved.We illustrate the new algorithm with results for the two-dimensional Ising model and propose a toy model to find the optimal parameter set for the parallel tempering routine within no time.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin-glass models where we find that a simple nearest-neighbor approach using a local n-fold-way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3D spin-glass model are very close and may even coincide.
The efficiency of parallel tempering Monte Carlo is studied for a two-dimensional Ising system of length L with N=L^2 spins. An external field is used to introduce a difference in free energy between the two low temperature states.\nIt is found that the number of replicas R_opt that optimizes the parallel tempering algorithm scales as the square root of the system size N. For two symmetric low temperature states, the time needed for equilibration is observed to grow as L^2.18. If a significant difference in free energy is present between the two states, this changes to L^1.02.\nIt is therefore established that parallel tempering is sped up by a factor of roughly L if an asymmetry is introduced between the low temperature states. This confirms previously made predictions for the efficiency of parallel tempering. These findings should be especially relevant when using parallel tempering for systems like spin glasses, where no information about the degeneracy of low temperature states is available prior to the simulation.
Ising machines have emerged as a potential technique for hardware acceleration in solving combinatorial optimization problems. However, an effective and hardware-implementation-friendly algorithm is still required to further enhance the fully-connected Ising machine's performance. Parallel momentum-tempering (PMT) algorithm is proposed in this paper, incorporating the full parallel updating capability of momentum annealing (MA) with the global exploration in the solution space provided by parallel tempering (PT). PMT exhibits an excellent performance, and a PMT with 8 replicas surpasses MA on the WK2000 max-cut benchmark, achieving a considerable 80%(1k/6k) reduction in required iteration steps while obtaining a mean cut value of 33,160. Besides, PMT is well-matched for hardware implementation versus MA, benefiting from invariant parameters in each replica.
Parametrized artificial neural networks (ANNs) can be very expressive ansatzes for variational algorithms, reaching state-of-the-art energies on many quantum many-body Hamiltonians. Nevertheless, the training of the ANN can be slow and stymied by the presence of local minima in the parameter landscape. One approach to mitigate this issue is to use parallel tempering methods, and in this work, we focus on the role played by the temperature distribution of the parallel tempering replicas. Using an adaptive method that adjusts the temperatures in order to equate the exchange probability between neighboring replicas, we show that this temperature optimization can significantly increase the success rate of the variational algorithm with negligible computational cost by eliminating bottlenecks in the replicas' random walk. We demonstrate this using two different neural networks, a restricted Boltzmann machine and a feedforward network, which we use to study a toy problem based on a permutation invariant Hamiltonian with a pernicious local minimum and the J_{1}-J_{2} model on a rectangular lattice.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.