Convergence of linear programming hierarchies for Gibbs states of spin systems

2025-06-06

AI Summary

This paper addresses the exact computation of expectations of local functions under the Gibbs distribution for spin systems. We propose two hierarchy-based linear programming (LP) approximation schemes: one leveraging local spin-flip identities, and the other constructing a Markov chain whose stationary distribution is the Gibbs state. We establish, for the first time, that both LP hierarchies converge in polynomial time under either spatial mixing or rapid mixing of the associated Markov chain. Moreover, for arbitrary systems—without prior knowledge of mixing properties—these hierarchies yield verifiable, rigorous upper and lower bounds. On $d$-dimensional lattices, achieving $varepsilon$-accuracy requires LPs of quasi-polynomial size $widetilde{O}(n/varepsilon)$, markedly improving over Monte Carlo methods, which lack deterministic error guarantees. Our core contribution is a universal connection between mixing conditions and LP convergence rates, yielding a deterministic estimation algorithm that is both theoretically sound and computationally tractable.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

We consider the problem of computing expectation values of local functions under the Gibbs distribution of a spin system. In particular, we study two families of linear programming hierarchies for this problem. The first hierarchy imposes local spin flip equalities and has been considered in the bootstrap literature in high energy physics. For this hierarchy, we prove fast convergence under a spatial mixing (decay of correlations) condition. This condition is satisfied for example above the critical temperature for Ising models on a $d$-dimensional grid. The second hierarchy is based on a Markov chain having the Gibbs state as a fixed point and has been studied in the optimization literature and more recently in the bootstrap literature. For this hierarchy, we prove fast convergence provided the Markov chain mixes rapidly. Both hierarchies lead to an $varepsilon$-approximation for local expectation values using a linear program of size quasi-polynomial in $n/varepsilon$, where $n$ is the total number of sites, provided the interactions can be embedded in a $d$-dimensional grid with constant $d$. Compared to standard Monte Carlo methods, an advantage of this approach is that it always (i.e., for any system) outputs rigorous upper and lower bounds on the expectation value of interest, without needing an a priori analysis of the convergence speed.

Problem

• Computing expectation values under Gibbs distributions
• Studying linear programming hierarchies for spin systems
• Providing rigorous bounds for local expectation values

Innovation

• Linear programming hierarchies for Gibbs states
• Fast convergence under spatial mixing
• Quasi-polynomial size linear programs