CoRMF: Criticality-Ordered Recurrent Mean Field Ising Solver
Zhenyu Pan, Ammar Gilani, En-Jui Kuo, Zhuo Liu
TL;DR
CoRMF presents a neural-solver for forward Ising inference by combining a criticality-ordered autoregressive factorization with an RNN to form a variational mean-field model. The method trains in a self-guided fashion using a variance-reduced reinforcement-style gradient and provides a mathematically grounded bound on the approximation error via matrix cut decomposition. Empirically, CoRMF demonstrates improved variational energy estimates and magnetization across several Ising datasets, highlighting its scalability and robustness when the edge-importance ordering is well-conditioned. The work offers a generic, data-free variational framework that can be applied to diverse forward Ising problems with reduced parameterization and theoretical guarantees, while outlining avenues for improving ordering strategies and extending to broader graphical models.
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.