CoRMF: Criticality-Ordered Recurrent Mean Field Ising Solver

TL;DR

CoRMF introduces a novel RNN-based solver for Ising models that leverages criticality ordering and mean-field approximation, enabling efficient probabilistic inference on complex graphs with theoretical error bounds.

Contribution

The paper presents a new criticality-ordered RNN approach that unifies variational mean-field and RNN methods for efficient Ising model inference, with theoretical and empirical validation.

Findings

Achieves tighter error bounds than naive mean-field.

Effectively solves Ising problems without data or evidence.

Demonstrates utility on various Ising datasets.

Abstract

We propose an RNN-based efficient Ising model solver, the Criticality-ordered Recurrent Mean Field (CoRMF), for forward Ising problems. In its core, a criticality-ordered spin sequence of an $N$-spin Ising model is introduced by sorting mission-critical edges with greedy algorithm, such that an autoregressive mean-field factorization can be utilized and optimized with Recurrent Neural Networks (RNNs). Our method has two notable characteristics: (i) by leveraging the approximated tree structure of the underlying Ising graph, the newly-obtained criticality order enables the unification between variational mean-field and RNN, allowing the generally intractable Ising model to be efficiently probed with probabilistic inference; (ii) it is well-modulized, model-independent while at the same time expressive enough, and hence fully applicable to any forward Ising inference problems with minimal effort. Computationally, by using a variance-reduced Monte Carlo gradient estimator, CoRFM solves the Ising problems in a self-train fashion without data/evidence, and the inference tasks can be executed by directly sampling from RNN. Theoretically, we establish a provably tighter error bound than naive mean-field by using the matrix cut decomposition machineries. Numerically, we demonstrate the utility of this framework on a series of Ising datasets.