Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning the Inverse Temperature of Ising Models under Hard Constraints using One Sample

Rohan Chauhan, Ioannis Panageas

TL;DR

The paper develops a near-linear-time method to estimate the inverse temperature $\beta$ of a truncated Ising model from a single sample, where the truncation set $S$ is the solution space of a bounded-degree $k$-SAT formula. It relies on a maximum pseudolikelihood estimator, and its consistency is proven by bounding the first and second derivatives of the pseudo-likelihood: a controlled gradient at $\beta^*$ and a strongly convex objective with parameter $\Omega(n/\Delta^3)$, shown via an independent-set conditioning argument and a Lovász Local Lemma construction that guarantees many one-flip neighbors within $S$. The analysis combines exchangeable-pair techniques for concentration with intricate combinatorial arguments (LLL, disjoint neighborhoods, and magnetization bounds) to bound the Hessian from below, enabling a projected gradient-descent algorithm running in $\mathcal{O}(\Delta^3 n \log n)$ time that achieves $|\hat{\beta}-\beta^*| \le c\Delta^3/\sqrt{n}$ with high probability, for $k \ge \Omega(\Delta^3(1+\log(d^2k+1)))$. These results extend single-sample learning to hard-constrained Ising models and provide a quantitative bridge between truncation structure and statistical-optimization guarantees, with implications for parameter inference in complex, non-product graphical models shaped by logical constraints.

Abstract

We consider the problem of estimating inverse temperature parameter $β$ of an $n$-dimensional truncated Ising model using a single sample. Given a graph $G = (V,E)$ with $n$ vertices, a truncated Ising model is a probability distribution over the $n$-dimensional hypercube $\{-1,1\}^n$ where each configuration $\mathbfσ$ is constrained to lie in a truncation set $S \subseteq \{-1,1\}^n$ and has probability $\Pr(\mathbfσ) \propto \exp(β\mathbfσ^\top A\mathbfσ)$ with $A$ being the adjacency matrix of $G$. We adopt the recent setting of [Galanis et al. SODA'24], where the truncation set $S$ can be expressed as the set of satisfying assignments of a $k$-SAT formula. Given a single sample $\mathbfσ$ from a truncated Ising model, with inverse parameter $β^*$, underlying graph $G$ of bounded degree $Δ$ and $S$ being expressed as the set of satisfying assignments of a $k$-SAT formula, we design in nearly $O(n)$ time an estimator $\hatβ$ that is $O(Δ^3/\sqrt{n})$-consistent with the true parameter $β^*$ for $k \gtrsim \log(d^2k)Δ^3.$ Our estimator is based on the maximization of the pseudolikelihood, a notion that has received extensive analysis for various probabilistic models without [Chatterjee, Annals of Statistics '07] or with truncation [Galanis et al. SODA '24]. Our approach generalizes recent techniques from [Daskalakis et al. STOC '19, Galanis et al. SODA '24], to confront the more challenging setting of the truncated Ising model.

Learning the Inverse Temperature of Ising Models under Hard Constraints using One Sample

TL;DR

The paper develops a near-linear-time method to estimate the inverse temperature of a truncated Ising model from a single sample, where the truncation set is the solution space of a bounded-degree -SAT formula. It relies on a maximum pseudolikelihood estimator, and its consistency is proven by bounding the first and second derivatives of the pseudo-likelihood: a controlled gradient at and a strongly convex objective with parameter , shown via an independent-set conditioning argument and a Lovász Local Lemma construction that guarantees many one-flip neighbors within . The analysis combines exchangeable-pair techniques for concentration with intricate combinatorial arguments (LLL, disjoint neighborhoods, and magnetization bounds) to bound the Hessian from below, enabling a projected gradient-descent algorithm running in time that achieves with high probability, for . These results extend single-sample learning to hard-constrained Ising models and provide a quantitative bridge between truncation structure and statistical-optimization guarantees, with implications for parameter inference in complex, non-product graphical models shaped by logical constraints.

Abstract

We consider the problem of estimating inverse temperature parameter of an -dimensional truncated Ising model using a single sample. Given a graph with vertices, a truncated Ising model is a probability distribution over the -dimensional hypercube where each configuration is constrained to lie in a truncation set and has probability with being the adjacency matrix of . We adopt the recent setting of [Galanis et al. SODA'24], where the truncation set can be expressed as the set of satisfying assignments of a -SAT formula. Given a single sample from a truncated Ising model, with inverse parameter , underlying graph of bounded degree and being expressed as the set of satisfying assignments of a -SAT formula, we design in nearly time an estimator that is -consistent with the true parameter for Our estimator is based on the maximization of the pseudolikelihood, a notion that has received extensive analysis for various probabilistic models without [Chatterjee, Annals of Statistics '07] or with truncation [Galanis et al. SODA '24]. Our approach generalizes recent techniques from [Daskalakis et al. STOC '19, Galanis et al. SODA '24], to confront the more challenging setting of the truncated Ising model.

Paper Structure

This paper contains 28 sections, 17 theorems, 99 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. Related Work
  5. Preliminaries
  6. Notation
  7. Maximum Pseudo-Likelihood Estimation
  8. An Auxiliary Lemma
  9. Learning Truncated Ising Models
  10. Roadmap for proving Theorem \ref{['thm:main']}
  11. Analysis of First Moment
  12. Second Derivative Bound
  13. Ensuring Flippability
  14. Bounding the Magnetizations
  15. Establishing Lemma \ref{['lem:hess_mean']}
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\bm{\sigma}$ be a single sample from a truncated, $n$-dimensional Ising model with inverse temperature $\beta^*$, where the truncation set is captured by the satisfying assignments of a $k$-SAT formula $\Phi_{n,k,d}$ and the underlying graph $G$ has maximum degree $\Delta$ of order $o(n^{1/6})$

Figures (1)

  • Figure 1: Two typical spin configurations over the Ising model at temperatures $\beta_1, \beta_2$ with $\beta_1 \gg \beta_2$ (equivalently at a lower temperature $T_1$ and a higher temperature $T_2$ where $T \propto 1/\beta$). Each node has spin $+1$ or spin $-1$. The left panel shows the configuration at a lower temperature, with alignment producing large domains of positive and negative spin assignments, while the right panel exhibits a more disordered pattern.

Theorems & Definitions (27)

  • Theorem 1: Informal Version of Theorem \ref{['thm:main']}
  • Remark : Consistency
  • Remark : Lower Bounds
  • Lemma 2.1: Symmetric Lovász Local Lemma
  • Remark : Assumptions
  • Theorem 2: Main result
  • Remark : The algorithm
  • Lemma 3.1
  • Lemma 3.2: Upper Bound on $\phi_1(\beta; \bm{\sigma})$ in Probability
  • Lemma 3.3: Lower Bound on $\phi_2(\beta; \bm{\sigma})$ in Probability
  • ...and 17 more