  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.
