We developed Graemlin 2.0, a new multiple network aligner with (1) a new multi-stage approach to local network alignment; (2) a novel scoring function that can use arbitrary features of a multiple network alignment, such as protein deletions, protein duplications, protein mutations, and interaction losses; (3) a parameter learning algorithm that uses a training set of known network alignments to learn parameters for our scoring function and thereby adapt it to any set of networks; and (4) an algorithm that uses our scoring function to find approximate multiple network alignments in linear time. We tested Graemlin 2.0's accuracy on protein interaction networks from IntAct, DIP, and the Stanford Network Database. We show that, on each of these datasets, Graemlin 2.0 has higher sensitivity and specificity than existing network aligners. Graemlin 2.0 is available under the GNU public license at http://graemlin.stanford.edu .