Unbiased Monte Carlo cluster updates with autoregressive neural networks

Efficient sampling of complex high-dimensional probability distributions is a central task in computational science. Machine learning methods like autoregressive neural networks, used with Markov chain Monte Carlo sampling, provide good approximations to such distributions but suffer from either intrinsic bias or high variance. In this Letter, we propose a way to make this approximation unbiased and with low variance. Our method uses physical symmetries and variable-size cluster updates which utilize the structure of autoregressive factorization. We test our method for first- and second-order phase transitions of classical spin systems, showing its viability for critical systems and in the presence of metastable states.

Figure 1

Example of three steps of neural cluster updates with symmetries (NCUSs) applied to a $4\times 4$ spin model. The columns correspond to different lines in Alg. 1. The last $k$ spins that can be flipped are highlighted in $\mathrm{blue}$. If a proposal is accepted, the spins actually flipped are shown in $\mathrm{red}$. For translations, the original borders of the lattice are shown in $\mathrm{green}$. For reflections, the plane of reflection is shown in $\mathrm{yellow}$, and yellow borders around the lattice indicate a reflection along the $z$ axis (across the $xy$ plane).

Figure 2

(a) Integrated autocorrelation time $\tau$ as a function of temperature on the $16\times 16$ Ising model. For neural importance sampling (NIS), we use the increased variance from the reweighting procedure as the effective autocorrelation time. The inset focuses on their behaviors near the critical point and uses the logarithmic scale on the $y$ axis. (b) Autocorrelation functions $r\left(t\right)$ on the $16\times 16$ Ising model at $\beta =0.44$. The inset uses the logarithmic scale on the $x$ axis to focus on their behaviors at small $t$.

Figure 3

(a) Sketch of the expected phase diagram of the frustrated plaquette model (FPM) for $J_1=J_3=-1$ with example ground states in the ferromagnetic (FM) and the ferrimagnetic (fM) phases. The dashed vertical line represents the temperature cut we numerically study. (b) Energy per site $\varepsilon$ and (c) spontaneous magnetization per site $m$ of the FPM for $J_1=J_3=-1$, $K=2$ as functions of temperature with lattice sizes up to $L=32$, obtained by neural cluster updates with symmetries (NCUSs). The dashed vertical line indicates the estimated transition point ${\beta }_c$.

Figure 4

(a) Integrated autocorrelation time $\tau$ as a function of temperature on the $32\times 32$ frustrated plaquette model (FPM). For neural importance sampling (NIS), we use the increased variance from the reweighting procedure as the effective autocorrelation time. The inset focuses on their behaviors near the critical point and uses the logarithmic scale on the $y$ axis. (b) Autocorrelation functions $r\left(t\right)$ on the $32\times 32$ FPM at $\beta =0.2$. The inset uses the logarithmic scale on the $x$ axis to focus on their behaviors at small $t$.

Figure 5

Probability distribution of the energy per site $p\left(\varepsilon \right)$ for the frustrated plaquette model (FPM) with different lattice sizes $L$ at their respective phase transition temperatures, obtained by neural cluster updates with symmetries (NCUSs).