[Paper Review] Machine-learning-assisted Monte Carlo fails at sampling computationally hard problems

Simone Ciarella, Jeanne Trinquier+2 more|arXiv (Cornell University)|Oct 20, 2022

Statistical Methods and Inference4 citations

TL;DR

This paper investigates whether machine-learning-assisted Monte Carlo methods can efficiently sample computationally hard problems, focusing on the antiferromagnetic Potts model on random graphs—equivalent to graph coloring at zero temperature. Despite testing multiple architectures, including autoregressive models and variational methods, all approaches fail due to mode collapse and poor acceptance rates in global MCMC moves, establishing this class of problems as a robust benchmark for future sampling algorithms.

ABSTRACT

Several strategies have been recently proposed in order to improve Monte Carlo sampling efficiency using machine learning tools. Here, we challenge these methods by considering a class of problems that are known to be exponentially hard to sample using conventional local Monte Carlo at low enough temperatures. In particular, we study the antiferromagnetic Potts model on a random graph, which reduces to the coloring of random graphs at zero temperature. We test several machine-learning-assisted Monte Carlo approaches, and we find that they all fail. Our work thus provides good benchmarks for future proposals for smart sampling algorithms.

Motivation & Objective

• To evaluate whether machine learning can universally accelerate Monte Carlo sampling in computationally hard systems.
• To test if ML-assisted MCMC can overcome exponential slowdowns in glassy systems with rough energy landscapes.
• To establish the antiferromagnetic Potts model on random graphs as a rigorous benchmark for sampling algorithms.
• To investigate the causes of failure in ML-assisted sampling, including mode collapse and poor acceptance rates.
• To assess the role of symmetry and model expressivity in the failure of variational and autoregressive approaches.

Proposed method

• The study uses the antiferromagnetic Potts model on random graphs as a prototypical hard-to-sample system, equivalent to graph coloring at zero temperature.
• Multiple ML-assisted MCMC strategies are tested, including autoregressive models, variational inference, and sequential tempering with learned proposals.
• The Kullback-Leibler divergence between target and auxiliary distributions is used to quantify approximation quality, with a focus on its scaling with system size N.
• Training involves large ensembles of configurations sampled via standard MCMC, with architectures ranging from shallow to deep autoregressive models.
• The performance of global MCMC moves is evaluated by measuring acceptance rates and relaxation times in the presence of glassy dynamics.
• Symmetry-breaking via small local fields is introduced to test whether degeneracy hinders learning, with no improvement observed.

Experimental results

Research questions

• RQ1 Can machine learning provide a universal speedup for Monte Carlo sampling in systems with rough energy landscapes?
• RQ2 Why do autoregressive and variational models fail to represent the true Boltzmann distribution in hard glassy systems?
• RQ3 Does the presence of color symmetry or degeneracy in the solution space prevent effective learning of the auxiliary distribution?
• RQ4 Can regularization or increased model expressivity resolve the failure of ML-assisted MCMC in hard sampling problems?
• RQ5 Is the failure due to mode collapse, poor proposal quality, or intrinsic limitations of the learning framework?

Key findings

• All tested ML-assisted MCMC methods fail to sample the antiferromagnetic Potts model on random graphs at low temperatures, despite using various architectures and training schemes.
• The variational model exhibits mode collapse, learning only a subset of the degenerate ground states and failing to represent the true equilibrium distribution.
• The sequential tempering approach shows poor acceptance rates for global moves, with relaxation times increasing during evolution, indicating aging behavior typical of glassy systems.
• Increasing model expressivity leads to overfitting (low entropy), while regularization causes underfitting (high entropy), both degrading MCMC efficiency.
• Adding small local fields to break color symmetry does not resolve the failure, indicating the core issue lies in the complexity of the glassy landscape, not symmetry degeneracy.
• The study establishes the random graph coloring problem as a robust benchmark for future sampling algorithms, highlighting its intractability even with ML assistance.