Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely, the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.

Figure 1

Illustration of a neural autoregressive distribution estimator. Arrows connected by blue segments correspond to connections with shared parameters.

Figure 2

Autocorrelation function $c\left(\tau \right)$ of the configuration energies $H$ obtained from single spin-flip MCMC simulations at inverse temperatures $\beta =0.1,0.2,\cdots ,1$ (bottom to top). Dashed lines represent stretched-exponential fitting functions.

Figure 3

Average of the negative log-likelihood $\left(\mathrm{nl}\right)$ as a function of the number of epochs $\left(N_E\right)$. Different curves correspond to different numbers of hidden units $N_H$, increasing from top to bottom. The size of the two-dimensional spin glass is $N=100$ and the inverse temperature is $\beta =1$.

Figure 4

Average energy per spin $E/N$ as a function of the number of hidden units $N_H$. The system size is $N=100$ and the inverse temperature is $\beta =1$. The black squares correspond to the expectation value over the probability distribution learned by the NADE [see Eq. (8)]. The red dots correspond to the results of the NADE-driven MCMC simulation. The horizontal blue line indicates the statistically unbiased result corresponding to the Boltzmann distribution.

Figure 5

Histogram of ${10}^5$ sampled configuration energies per spin $H/N$. The full yellow columns correspond to a single spin-flip MCMC simulation $(8\times {10}^7$ steps, every $800\mathrm{th}$ configuration is counted), the empty black columns to a NADE-driven MCMC simulation (every $10\mathrm{th}$ configuration is counted), and the empty red columns to the energies sampled via ancestral sampling from the probability distribution learned by the NADE. $N=100,\beta =1$, and $N_H=64$.

Figure 6

Relative error $(E_{\mathrm{es}}-E)/\left|E\right|$ of the estimated energy $E_{\mathrm{es}}$ with respect to the exact value $E$, as a function of the system size $N$. For the black squares $E_{\mathrm{es}}$ is the expectation value over the probability distribution learned by NADE, while for the red dots $E_{\mathrm{es}}$ is the result of a NADE-driven MCMC simulation. $N_H=64$ and $\beta =1$.

Figure 7

Percentage acceptance ratio $A_R$ in NADE-driven MCMC simulations, as a function of hidden-neuron number $N_H$. $N=100$ and $\beta =1$.

Figure 8

Percentage acceptance ratio $A_R$ in NADE-driven MCMC simulations for 10 instances of the spin-glass Hamiltonian with different realizations of the gaussian random couplings. The red dashed line indicates the average value. $N=100,\beta =1$, and $N_H=64$.

Figure 9

Percentage acceptance ratio $A_R$ in NADE-driven MCMC simulations as a function of the system size $N$. $\beta =1$ and $N_H=64$.

Figure 10

Percentage acceptance ratio $A_R$ in the NADE-driven MCMC simulations of the sequential tempering procedure, as a function of the inverse temperature $\beta$. $N=100$ and $N_H=64$.

Figure 11

Autocorrelation function $c\left(\tau \right)$ of the configuration energies at $\beta =2$. The red line corresponds to the single spin-flip MCMC algorithm (data averaged of 25 simulations started from different initial configurations), while the black dashed line corresponds to the last NADE-driven MCMC simulation of the sequential tempering procedure. $N=100$ and $N_H=64$.

Figure 12

Histogram of ${10}^5$ configuration energies per spin $H/N$ generated in the last NADE-driven MCMC simulation of the sequential tempering procedure at ${\beta }_{\max }=2$. Every $10\mathrm{th}$ configuration is counted. The 5 datasets correspond to just as many sequences started from different initial configurations at ${\beta }_0=0.5$, for the same realization of the spin-glass model. $N=100$ and $N_H=64$. The blue vertical dashed line indicates the ground-state energy for this instance of the spin-glass model, obtained from the spin-glass server [54].

Figure 13

Histogram of ${10}^5$ configuration energies per spin $H/N$ sampled at $\beta =2$. The empty blue columns correspond to the single spin-flip MCMC algorithm (data averaged over 25 simulations started from different configurations, ran for $8\times {10}^7$ steps, every $800\mathrm{th}$ configuration is counted). The gray columns correspond to the last NADE-driven MCMC simulation of the sequential tempering procedure (every $10\mathrm{th}$ configuration is counted). For comparison, the full red columns indicate the first ${10}^5$ configurations sampled via the single spin-flip MCMC algorithm. The vertical dashed line indicates the ground-state energy, obtained from the spin-glass server [54]. $N=100$ and $N_H=64$.

Figure 14

Configuration energy per spin $H/N$ as a function of the number of MCMC steps $t$, at $\beta =2$. Dashed black line corresponds to the last NADE-driven MCMC simulation of the sequential tempering procedure. Solid red line corresponds to the single spin-flip MCMC simulation. The horizontal blue line indicates the ground-state energy, obtained from the spin-glass server [54]. $N=100$ and $N_H=64$.