Computationally sufficient statistics for Ising models

2026-02-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research addresses the computational challenge of learning Gibbs distributions, specifically Ising models, when only limited statistical observations are available, rather than full sample configurations. Traditionally, learning these models from sufficient statistics is computationally hard, while efficient algorithms typically require complete sample access. The authors demonstrate that it is feasible to reconstruct Ising model parameters, including couplings and magnetic fields, for models with an $\ell_{1}$-width $\gamma$ by observing statistics up to an order of $O(\gamma)$. Their approach modifies the Interaction Screening Estimator (ISE) by approximating its gradient with a polynomial, allowing for convex optimization with a corrupted gradient oracle. This method requires $O(e^{8\gamma}\gamma^{5}\log(p))$ observations to achieve learning guarantees comparable to those using full configurations. The study also explores scenarios where prior structural information, such as a known $D$-regular graph, can further reduce the required observational power to statistics up to order $D+1$, with $n=O(2^{2D}e^{8\gamma}\gamma^{4})$ observations.

Key takeaway

For AI scientists working with complex physical systems where full sample configurations are impractical, this research offers a pathway to efficiently learn Ising model parameters. You should consider implementing gradient approximation techniques based on $O(\gamma)$ order statistics to infer model structure and couplings. This approach can significantly reduce data requirements, especially if you have prior knowledge about the model's graphical structure, enabling tractable learning from limited observational data.

Key insights

Learning Ising models from limited statistics is feasible by approximating gradients with polynomials up to $O(\gamma)$ order.

Principles

• Observational power interacts with computational tractability.
• Prior structural information can reduce required statistics order.

Method

The method approximates the Interaction Screening (IS) loss gradient using a polynomial, treating it as convex optimization with a corrupted gradient oracle. It sequentially learns second-order terms, structure, and linear terms.

In practice

• Reconstruct Ising model parameters using $O(\gamma)$ order statistics.
• Apply Interaction Screening Estimator with polynomial gradient approximation.

Topics

• Ising Models
• Gibbs Distributions
• Sufficient Statistics
• Interaction Screening
• Parameter Learning

Best for: AI Scientist, AI Researcher, Research Scientist, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.