Structure Learning in Inverse Ising Problems Using $\ell_2$-Regularized Linear Estimator
This addresses the problem of inferring network structures in statistical physics and machine learning, offering a method to handle model mismatch, though it is incremental as it builds on existing pseudolikelihood and regularization techniques.
The paper tackles the inverse Ising problem under model mismatch, showing that naive linear regression can perfectly identify network structure when the number of spins is less than the dataset size, but fails in underdetermined cases; a two-stage estimator with ridge regression and pruning is proposed to achieve perfect identification even with fewer data points.
The inference performance of the pseudolikelihood method is discussed in the framework of the inverse Ising problem when the $\ell_2$-regularized (ridge) linear regression is adopted. This setup is introduced for theoretically investigating the situation where the data generation model is different from the inference one, namely the model mismatch situation. In the teacher-student scenario under the assumption that the teacher couplings are sparse, the analysis is conducted using the replica and cavity methods, with a special focus on whether the presence/absence of teacher couplings is correctly inferred or not. The result indicates that despite the model mismatch, one can perfectly identify the network structure using naive linear regression without regularization when the number of spins $N$ is smaller than the dataset size $M$, in the thermodynamic limit $N\to \infty$. Further, to access the underdetermined region $M < N$, we examine the effect of the $\ell_2$ regularization, and find that biases appear in all the coupling estimates, preventing the perfect identification of the network structure. We, however, find that the biases are shown to decay exponentially fast as the distance from the center spin chosen in the pseudolikelihood method grows. Based on this finding, we propose a two-stage estimator: In the first stage, the ridge regression is used and the estimates are pruned by a relatively small threshold; in the second stage the naive linear regression is conducted only on the remaining couplings, and the resultant estimates are again pruned by another relatively large threshold. This estimator with the appropriate regularization coefficient and thresholds is shown to achieve the perfect identification of the network structure even in $0<M/N<1$. Results of extensive numerical experiments support these findings.