Machine-learning-assisted Monte Carlo fails at sampling computationally hard problems

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.

Learning nonequilibrium statistical mechanics and dynamical phase transitions

“…This important sampling is on the configurations from a distribution at each time point, different from the sampling on trajectories, which may be harder to sample as the trajectory space grows exponentially with time points. Besides, learning the multiple probability peaks can be affected by the mode-collapse: For rugged distributions, not all modes of the target distribution may be directly captured by the VAN 47 . To alleviate the mode-collapse, besides the importance sampling used here, temperature annealing 19 and variational annealing 48 can be employed.…”

The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems

“…• The one parameter (1P) architecture, where a single weight parameter is multiplied by the sums of the input variables, and then the sigma function is applied. This architecture was already used for the CW system in [35]. The total number of parameters scales as N .…”

“…It can be seen as the worst performance of the 1-layer networks (1P, 1L) is due to the difficulty of correctly representing the configurations with magnetization different than zero in the proximity of the phase transition. This could be due to mode collapse problems [35], which do not affect the deeper AR-NN architectures tested.…”

“…In this work, while the ARNN architecture of an SK model was derived, its learnability was not thoroughly examined. The problem of finding the configurations of minimum energy for the SK model is known to belong to the NP-hard class, and the effectiveness of the ARNN approach in solving this problem is still uncertain and a matter of ongoing research [27,34,35]. Further systematic studies are needed to fully understand the learnability of the ARNN architecture presented in this work at very low temperatures and also on different systems.…”

“…Then similar approaches have been used in different contexts, and domains of classical [23][24][25][26][27] and quantum statistical physics [28][29][30][31][32][33]. The ability of the ARNNs to efficiently generate samples, thanks to the ancestral sampling procedure, opened the way to overcome the slowdown of Monte-Carlo methods for frustrated or complex systems, although two recent works questioned the real gain in very frustrated systems [34,35]. The use of ARNNs in statistical physics problems has largely relied on pre-existing neural network architectures which may not be well-suited for the particular problem at hand.…”