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ShareWe discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are......
We discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are considered: (1) the +/- J spin glass, and (2) the Gaussian spin glass. We also study a smooth transition between these two extremal cases. The computational complexity of all studied spin glass systems is found to be dominated by rare events of extremely hard spin glass samples. We show that complexity of all studied spin glass systems is closely related to Frechet extremal value distribution. In a hybrid algorithm that combines the hierarchical Bayesian optimization algorithm (hBOA) with a deterministic bit-flip hill climber, the number of steps performed by both the global searcher (hBOA) and the local searcher follow Frechet distributions. Nonetheless, unlike in methods based purely on local search, the parameters of these distributions confirm good scalability of hBOA with local search. We further argue that standard performance measures for optimization algorithms---such as the average number of evaluations until convergence---can be misleading. Finally, our results indicate that for highly multimodal constraint satisfaction problems, such as Ising spin glasses, recombination-based search can provide qualitatively better results than mutation-based search.
hBOA actually outperforms even state of the art methods based on small random perturbations used typically in statistical physics on this problem like flat histogram Markov chain Monte Carlo, the complexity of which grows exponentially. Spin glasses are tough for all algorithms that make small randomized changes in solutions and are not capable of changing blocks of spins; this is because of their properties---there are energy barriers, many good local optima are relatively far from local optima, etc. This was investigated before in e.g. Dayal et al. (2004) who showed that flat histogram MCMC takes exponential time. Also, SA won't do better than flat histogram MCMC. There is more on this in the paper associated with this presenttion:
http://arxiv.org/abs/cs.NE/0402030
There is also an extended version of the paper which I can provide if this is not sufficient.
Of course there is more to say to this, and for example cluster exact approximation works very well on spin glasses, and can in fact be combined with hBOA (http://medal.cs.umsl.edu/files/2006002. pdf). But that's a very different method than the usual SA techniques (it's based on maximum flow), it changes large domains of spins optimally in each step. If you want more pointers, I can provide :-)
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