Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Neng Huang, Will Perkins, Aaron Potechin

TL;DR

The paper demonstrates a fundamental gap between search and sampling for the anti-ferromagnetic Ising model on random graphs: although near-optimal search is tractable via mean-field-like methods, stable sampling of the Gibbs distribution becomes provably hard when the inverse temperature exceeds the condensation threshold, as $d$ grows. The authors extend the interpolation framework between the Sherrington–Kirkpatrick (SK) model and diluted sparse graphs to include average energy and overlap, not just free energy, and they employ a Poisson Stein-Chen approach to mirror Gaussian integration by parts in the sparse setting. They establish that the sparse model inherits SK-like overlap structure in the high-degree limit and that disorder chaos transfers from SK to the sparse model, yielding a constant Wasserstein-distance gap for any stable sampler at $eta>1$. This combination of energy/overlap interpolation and disorder chaos provides a robust mechanism to rule out stable sampling algorithms and highlights a sharp separation between optimization and sampling in dilute mean-field systems.

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 $β>1$ (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.

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

TL;DR

The paper demonstrates a fundamental gap between search and sampling for the anti-ferromagnetic Ising model on random graphs: although near-optimal search is tractable via mean-field-like methods, stable sampling of the Gibbs distribution becomes provably hard when the inverse temperature exceeds the condensation threshold, as grows. The authors extend the interpolation framework between the Sherrington–Kirkpatrick (SK) model and diluted sparse graphs to include average energy and overlap, not just free energy, and they employ a Poisson Stein-Chen approach to mirror Gaussian integration by parts in the sparse setting. They establish that the sparse model inherits SK-like overlap structure in the high-degree limit and that disorder chaos transfers from SK to the sparse model, yielding a constant Wasserstein-distance gap for any stable sampler at . This combination of energy/overlap interpolation and disorder chaos provides a robust mechanism to rule out stable sampling algorithms and highlights a sharp separation between optimization and sampling in dilute mean-field systems.

Abstract

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree for large constant , proving that when the normalized inverse temperature satisfies (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 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 for large . 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.
Paper Structure (16 sections, 35 theorems, 144 equations)

This paper contains 16 sections, 35 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Techniques
  4. Organization
  5. Overview of the proof
  6. Gibbs distributions
  7. Gibbs average
  8. Free energy
  9. Correspondence between dense and sparse models
  10. Disorder chaos and stable algorithms
  11. Interpolating the average energy between SK and the diluted model
  12. Interpolating the average overlap between SK and the diluted model
  13. Disorder chaos for sparse models
  14. Free energy of bisection models
  15. SK model with zero magnetization
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{\mathsf{ALG}_n\}_{n \geq 1}$ be a family of randomized sampling algorithms where $\mathsf{ALG}_n$ takes as input an $n$-vertex (multi-)graph $G$ and an inverse temperature parameter $\beta$ and produces an output law $\mu^{\mathsf{ALG}}_{\beta, G}$ over $\{-1, 1\}^n$. For every $\beta > 1$ an

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3: dembo_extremal_2017
  • Proposition 2.4: Restatement of Proposition \ref{['prop:average_energy_SK_sparse_sec1']}
  • Proposition 2.5: Restatement of Proposition \ref{['prop:overlap_interpolation_sec1']}
  • ...and 45 more