  The inverse Potts problem to infer a Boltzmann distribution for homologous
protein sequences from their single-site and pairwise amino acid frequencies
recently attracts a great deal of attention in the studies of protein structure
and evolution. We study regularization and learning methods and how to tune
regularization parameters to correctly infer interactions in Boltzmann machine
learning. Using $L_2$ regularization for fields, group $L_1$ for couplings is
shown to be very effective for sparse couplings in comparison with $L_2$ and
$L_1$. Two regularization parameters are tuned to yield equal values for both
the sample and ensemble averages of evolutionary energy. Both averages smoothly
change and converge, but their learning profiles are very different between
learning methods. The Adam method is modified to make stepsize proportional to
the gradient for sparse couplings. It is shown by first inferring interactions
from protein sequences and then from Monte Carlo samples that the fields and
couplings can be well recovered, but that recovering the pairwise correlations
in the resolution of a total energy is harder for the natural proteins than for
the protein-like sequences. Selective temperature for folding/structural
constrains in protein evolution is also estimated.
