# Unbiased Monte Carlo Cluster Updates with Autoregressive Neural Networks

arXiv: https://arxiv.org/abs/2105.05650 · HTML: https://mlsanity.com/paper/2105.05650/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 72 references that are in the map.
- First posted on arXiv 12 May 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Generalizing Hamiltonian Monte Carlo with Neural Networks](https://mlsanity.com/paper/1711.09268/) (2017)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [A-NICE-MC: Adversarial Training for MCMC](https://mlsanity.com/paper/1706.07561/) (2017)
- [Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines](https://mlsanity.com/paper/1610.02746/) (2016)
- [The Lov'asz Theta Function for Random Regular Graphs and Community Detection in the Hard Regime](https://mlsanity.com/paper/1705.01194/) (2017)

## Built on by (most cited first, 17 of 17)

- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems](https://mlsanity.com/paper/2302.08347/) (2023)
- [Message Passing Variational Autoregressive Network for Solving Intractable Ising Models](https://mlsanity.com/paper/2404.06225/) (2024)
- [IsingFormer: Augmenting Parallel Tempering With Learned Proposals](https://mlsanity.com/paper/2509.23043/) (2025)
- [Nonlocal Monte Carlo via Reinforcement Learning](https://mlsanity.com/paper/2508.10520/) (2025)
- [An Empirical Study of Quantum Dynamics as a Ground State Problem with Neural Quantum States](https://mlsanity.com/paper/2206.09241/) (2022)
- [Neural network quantum state with proximal optimization: a ground-state searching scheme based on variational Monte Carlo](https://mlsanity.com/paper/2210.16493/) (2022)
- [High-precision regressors for particle physics](https://mlsanity.com/paper/2302.00753/) (2023)
- [Locality-constrained autoregressive cum conditional normalizing flow for lattice field theory simulations](https://mlsanity.com/paper/2304.01798/) (2023)
- [Statistical Mechanics Calculations Using Variational Autoregressive Networks and Quantum Annealing](https://mlsanity.com/paper/2404.19274/) (2024)
- [A method for quantifying the generalization capabilities of generative models for solving Ising models](https://mlsanity.com/paper/2405.03435/) (2024)
- [Machine learning the Ising transition: A comparison between discriminative and generative approaches](https://mlsanity.com/paper/2411.19370/) (2024)
- [Enhanced Neural Quantum State via Annealed Gradient Descent](https://mlsanity.com/paper/2607.18865/) (2026)

## Abstract (arXiv)

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.