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We investigate the learning performance of the pseudolikelihood maximizationmethod for inverse Ising problems. In the teacher-student scenario under theassumption that the teacher's couplings are sparse and the student does notknow the graphical structure, the learning curve and order parameters areassessed in the typical case using the replica and cavity methods fromstatistical mechanics. Our formulation is also applicable to a certain class ofcost functions having locality; the standard likelihood does not belong to thatclass. The derived analytical formulas indicate that the perfect inference ofthe presence/absence of the teacher's couplings is possible in thethermodynamic limit taking the number of spins $N$ as infinity while keepingthe dataset size $M$ proportional to $N$, as long as $\alpha=M/N > 2$.Meanwhile, the formulas also show that the estimated coupling valuescorresponding to the truly existing ones in the teacher tend to beoverestimated in the absolute value, manifesting the presence of estimationbias. These results are considered to be exact in the thermodynamic limit onlocally tree-like networks, such as the regular random or Erd\H{o}s--R\'enyigraphs. Numerical simulation results fully support the theoretical predictions.Additional biases in the estimators on loopy graphs are also discussed.

Learning performance in inverse Ising problems with sparse teacher couplings