Flow-based generative models for Markov chain Monte Carlo in lattice field theory

A Markov chain update scheme using a machine-learned flow-based generative model is proposed for Monte Carlo sampling in lattice field theories. The generative model may be optimized (trained) to produce samples from a distribution approximating the desired Boltzmann distribution determined by the lattice action of the theory being studied. Training the model systematically improves autocorrelation times in the Markov chain, even in regions of parameter space where standard Markov chain Monte Carlo algorithms exhibit critical slowing down in producing decorrelated updates. Moreover, the model may be trained without existing samples from the desired distribution. The algorithm is compared with HMC and local Metropolis sampling for ${\varphi }^4$ theory in two dimensions.

Figure 1

In (a), a normalizing flow is shown transforming samples $z$ from a prior distribution $r(z)$ to samples $\varphi$ distributed according to ${\tilde{p}}_f(\varphi )$. The mapping $f^{-1}(z)$ is constructed by composing inverse coupling layers $g_i^{-1}$ as defined in Eq. (10) in terms of neural networks $s_i$ and $t_i$ and shown diagrammatically in (b). By optimizing the neural networks within each coupling layer, ${\tilde{p}}_f(\varphi )$ can be made to approximate a distribution of interest, $p(\varphi )$.

Figure 2

Histograms of length of consecutive runs of Metropolis rejections in ML models at both 50% and 70% mean acceptance. Also shown is the same statistic for Markov chains generated via HMC, where mean acceptance was tuned to 70%. The frequency of long runs of rejections is consistently reduced for models trained to reach higher average acceptance. The ML and HMC ensembles at 70% acceptance display very similar distributions of rejection streaks.

Figure 3

Zero-momentum Green’s functions evaluated for parameter set E5. Results computed using $10^6$ configurations from the HMC, local Metropolis, and ML ensembles are consistent within statistical errors. Error bars indicate 68% confidence intervals estimated using bootstrap resampling with bins of size 100.

Figure 4

Effective pole masses evaluated for parameter set E5, defined by the arccosh estimator given in the main text. Results computed using $10^6$ configurations from the HMC, local Metropolis, and ML ensembles are consistent within statistical errors. Error bars indicate 68% confidence intervals estimated using bootstrap resampling with bins of size 100.

Figure 5

Susceptibility (${\chi }_2$) and Ising energy ($E$) estimated on all ensembles. Results computed using $10^6$ configurations from the HMC, local Metropolis, and ML ensembles are consistent within statistical errors. Errors indicate 68% confidence intervals estimated using bootstrap resampling with bins of size 100.

Figure 6

Statistical error varying with the number of samples $N$ in two candidate observables, ${\chi }_2$ and $E$, for the HMC, local Metropolis, and ML ensembles. The red dashed line shows a $1/\sqrt{N}$ curve normalized by the average error estimate of the three approaches at $N=1000$. Central values were estimated as 68% confidence intervals on each observable by bootstrap resampling ensemble subsets of size $N$. Error bars indicate 68% confidence intervals estimated using an external bootstrap resampling step.

Figure 7

Scaling of integrated autocorrelation time with respect to lattice size for HMC, local Metropolis, and flow-based MCMC. In (c) the upper sets of points in blue correspond to models trained to a mean acceptance rate of 50%, while the lower sets of points in green correspond to models trained to a mean acceptance rate of 70%. Dashed red lines display power-law fits to $L=\{10,12,14\}$ with labels $L^z$ specifying the scaling. The HMC and local Metropolis methods demonstrate power-law growth of ${\tau }_{\mathrm{int}}$, while ${\tau }_{\mathrm{int}}$ for the flow-based MCMC is consistent with a constant in $L$ and decreases as mean acceptance rate increases. Dot-dashed blue and green lines for the flow-based ensembles display lower bounds in terms of mean acceptance rate based on Eq. (18). Error bars indicate 68% confidence intervals estimated by bootstrap resampling and error propagation.