Learning performance in inverse Ising problems with sparse teacher couplings

Abstract

We investigate the learning performance of the pseudolikelihood maximization method for inverse Ising problems. In the teacher-student scenario under the assumption that the teacher's couplings are sparse and the student does not know the graphical structure, the learning curve and order parameters are assessed in the typical case using the replica and cavity methods from statistical mechanics. Our formulation is also applicable to a certain class of cost functions having locality; the standard likelihood does not belong to that class. The derived analytical formulas indicate that the perfect inference of the presence/absence of the teacher's couplings is possible in the thermodynamic limit taking the number of spins $N$ as infinity while keeping the dataset size $M$ proportional to $N$, as long as $\alpha =M/N>2$. Meanwhile, the formulas also show that the estimated coupling values corresponding to the truly existing ones in the teacher tend to be overestimated in the absolute value, manifesting the presence of estimation bias. These results are considered to be exact in the thermodynamic limit on locally tree-like networks, such as the regular random or Erdős--Rényi graphs. Numerical simulation results fully support the theoretical predictions. Additional biases in the estimators on loopy graphs are also discussed.

arXiv comment29 pages, 8 figures

References in corpus (15)

• Weak pairwise correlations imply strongly correlated network states in a neural population
• High-dimensional Ising model selection using ${\ell }_1$-regularized logistic regression
• Searching for collective behavior in a network of real neurons
• Inverse statistical problems: from the inverse Ising problem to data science
• Inverse Ising inference using all the data
• Adaptive Cluster Expansion for Inferring Boltzmann Machines with Noisy Data
• Small-correlation expansions for the inverse Ising problem
• Mean Field Theory For Non-Equilibrium Network Reconstruction
• The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods
• On the criticality of inferred models
• Pseudolikelihood Decimation Algorithm Improving the Inference of the Interaction Network in a General Class of Ising Models
• Bethe-Peierls approximation and the inverse Ising model
• A statistical physics approach to learning curves for the Inverse Ising problem
• Learning of couplings for random asymmetric kinetic Ising models revisited: random correlation matrices and learning curves
• Statistical mechanics of the inverse Ising problem and the optimal objective function

Cited by in corpus (2)

• Structure Learning in Inverse Ising Problems Using ${\ell }_2$-Regularized Linear Estimator
• Ising Model Selection Using ${\ell }_1$-Regularized Linear Regression: A Statistical Mechanics Analysis