Ground states of 2D ±JIsing spin glasses obtained via stationary Fokker–Planck sampling

Abstract

We investigate the performance of the recently proposed stationary Fokker-Planck sampling method considering a combinatorial optimization problem from statistical physics. The algorithmic procedure relies upon the numerical solution of a linear second order differential equation that depends on a diffusion-like parameter D. We apply it to the problem of finding ground states of 2d Ising spin glasses for the ±J−Model. We consider square lattices with side length up to L = 24 with two different types of boundary conditions and compare the results to those obtained by exact methods. A particular value of D is found that yields an optimal performance of the algorithm. We compare this optimal value of D to a percolation transition, which occurs when studying the connected clusters of spins flipped by the algorithm. Nevertheless, even for moderate lattice sizes, the algorithm has more and more problems to find the exact ground states. This means that the approach, at least in its standard form, seems to be inferior to other approaches like parallel tempering.

Bayesian Inference Based on Stationary Fokker-Planck Sampling

“…This experimental result is important because it shows the value of SFP sampling for large-scale systems. In the context of global optimization, it appears that SFP should be further adapted to alleviate to some extent the curse of dimensionality suffered by any stochastic optimization method in order to be competitive with the current best algorithms (Melchert & Hartmann, 2008). However, Table 5 and Figure 10 suggest that for density estimation purposes, the correct estimation of large-dimensional densities by SFP sampling is a polynomial time computational procedure, with a total number of loss function evaluations that behave linearly.…”

“…This experimental result is important because it shows the value of SFP sampling for large scale systems. In the context of global optimization, it appears that SFP should be further adapted to alleviate at some extent the curse of dimensionality suffered by any stochastic optimization method in order to be competitive with the current best algorithms (Melchert & Hartmann, 2008). However, Table 5 and Fig.…”