  Using methods of statistical physics, we analyse the error of learning
couplings in large Ising models from independent data (the inverse Ising
problem). We concentrate on learning based on local cost functions, such as the
pseudo-likelihood method for which the couplings are inferred independently for
each spin. Assuming that the data are generated from a true Ising model, we
compute the reconstruction error of the couplings using a combination of the
replica method with the cavity approach for densely connected systems. We show
that an explicit estimator based on a quadratic cost function achieves minimal
reconstruction error, but requires the length of the true coupling vector as
prior knowledge. A simple mean field estimator of the couplings which does not
need such knowledge is asymptotically optimal, i.e. when the number of
observations is much large than the number of spins. Comparison of the theory
with numerical simulations shows excellent agreement for data generated from
two models with random couplings in the high temperature region: a model with
independent couplings (Sherrington-Kirkpatrick model), and a model where the
matrix of couplings has a Wishart distribution.
