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.

翻译：我们提出了一种基于RNN的高效伊辛模型求解器——临界有序递归平均场（CoRMF），用于解决正向伊辛问题。其核心是通过贪心算法对关键边进行排序，引入$N$自旋伊辛模型的临界有序自旋序列，从而能够利用并优化基于递归神经网络（RNN）的自回归平均场分解。该方法具有两个显著特征：（i）利用底层伊辛图的近似树结构，新获得的临界序实现了变分平均场与RNN的统一，使得通常难以处理的伊辛模型能够通过概率推断高效求解；（ii）该模型模块化良好、与模型无关且同时具有足够的表达能力，因此能以最小工作量完全适用于任何正向伊辛推断问题。在计算方面，通过使用方差缩减的蒙特卡洛梯度估计器，CoRFM以无数据/证据的自训练方式求解伊辛问题，推断任务可直接通过从RNN中采样执行。在理论方面，我们利用矩阵切割分解机制建立了比朴素平均场更严格的误差界。在数值方面，我们通过一系列伊辛数据集展示了该框架的实用性。