Fig. 2.

Replicated SA on the perceptron, comparison between the interacting version (i.e., which seeks regions of high solution density) and the noninteracting version (i.e., standard SA), at $\alpha =0.3$ patterns per synapse using $y=3$ replicas. With optimized annealing/scoping parameters, the minimum number of iterations required to find a solution scales exponentially with $N$ for the standard case and polynomially for the interacting case. Ten samples were tested for each value of $N$ (the same samples in both cases). The bars represent averages and standard deviations (taken in logarithmic scale) and the lines represent fits. The interacting case was fitted by a function $a{N}^b$ with $a\simeq 0.13$, $b\simeq 1.7$, and the noninteracting case was fitted by a function $a{N}^b{e}^{c{N}^d}$ with $a\simeq 0.2$, $b\simeq 1.5$, $c\simeq 6.6\cdot {10}^{-4}$, $d\simeq 1.1$. Data are not available for the noninteracting case at $N=6401$ because we could not solve any of the problems in a reasonable time (the extrapolated value according to the fit is $\sim 3\cdot {10}^9$). The two datasets are slightly shifted relative to each other for presentation purposes. All of the details are reported in SI Appendix.