Hardness of sampling for the anti-ferromagnetic Ising model on random graphs

TL;DR

This paper proves that for large average degree and high inverse temperature, sampling from the anti-ferromagnetic Ising model on random graphs is computationally hard, highlighting a gap between optimization and sampling algorithms.

Contribution

It establishes a new hardness result for sampling from the anti-ferromagnetic Ising model at high temperature, extending interpolation techniques to compare energies and overlaps.

Findings

Sampling is hard for large d and β>1

No stable sampling algorithms for low positive temperature bisection distributions

Efficient algorithms exist for finding near optimizers

Abstract

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree $d$ for large constant $d$, proving that when the normalized inverse temperature satisfies $\beta>1$ (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in $W_2$ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions. Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree $d$ for large $d$. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models.