Accelerated Monte Carlo simulations with restricted Boltzmann machines

Despite their exceptional flexibility and popularity, Monte Carlo methods often suffer from slow mixing times for challenging statistical physics problems. We present a general strategy to overcome this difficulty by adopting ideas and techniques from the machine learning community. We fit the unnormalized probability of the physical model to a feed-forward neural network and reinterpret the architecture as a restricted Boltzmann machine. Then, exploiting its feature detection ability, we utilize the restricted Boltzmann machine to propose efficient Monte Carlo updates to speed up the simulation of the original physical system. We implement these ideas for the Falicov-Kimball model and demonstrate an improved acceptance ratio and autocorrelation time near the phase transition point.

Figure 1

(a) The restricted Boltzmann machine is an energy-based model for binary stochastic visible and hidden variables. Their probability distribution follows the Boltzmann distribution with the energy function in Eq. (4). (b) Viewing the RBM as a feed-forward neural network which maps the visible variables to the free energy Eq. (5). The gray arrows represent the first term of Eq. (5). The red circles are hidden neurons with the softplus activation function, corresponding to the second term of Eq. (5). Fitting $F\left(\mathbf{x}\right)$ to the log probability of the physical models [e.g., Eq. (2)] determines the weights and biases of the RBM.

Figure 2

Fitting of the log probability of the RBM, Eq. (5), to the one of the Falicov-Kamball model, Eq. (2), on a $N=8^2$ square lattice. The parameters are $U/t=4, T/t=0.15$, and we use $M=100$ hidden variables. We offset the data so that the minimum value is at zero.

Figure 3

Connection weights $W_{ij}$ of the restricted Boltzmann machine trained for the Falicov-Kimball model (1) on a square lattice with $N=8^2$ sites at $U/t=4, T/t=0.15$. Each tile represents the connection weights of a hidden neuron. We scale the weights in the range of $0–1$ for visibility.

Figure 4

Two strategies of proposing Monte Carlo updates using the RBM. (a) The blocked Gibbs sampling. Given the visible variables $\mathbf{x}$, we sample the hidden variable $\mathbf{h}$ under the conditional probability $p\left(\mathbf{h}\right|\mathbf{x})$, Eq. (7), then sample the visible variables under the conditional probability $p\left({\mathbf{x}}^{\prime }\right|\mathbf{h})$, Eq. (8). (b) The blocked Gibbs sampling with an additional Metropolis step in between. It updates the hidden variable $\mathbf{h}\to {\mathbf{h}}^{\prime }$ according to the log probability of the hidden variables, Eq. (9).

Figure 5

(a) The acceptance ratio and (b) the total energy autocorrelation time of the Falicov-Kimball model on a $N=8^2$ square lattice. Blue squares denote results of local bit-flip updates. The yellow and red dots are using the RBM update schemes of Figs. 4 and 4, respectively. The critical temperature is at $T/t\approx 0.15$ [33, 34, 35]. The estimated physical observables agree within error bars for all sampling approaches.

Figure 6

Connection weights $W_{ij}$ of the RBM at $T/t=0.13$. The other parameters are the same as the Fig. 3 of the main texts.