Background: In a computed protein multiple sequence alignment, the coreness of a column is the fraction of its substitutions that are in so-called core columns of the gold-standard reference alignment of its proteins. In benchmark suites of protein reference alignments, the core columns of the reference alignment are those that can be confidently labeled as correct, usually due to all residues in the column being sufficiently close in the spatial superposition of the known three-dimensional structures of the proteins. Typically the accuracy of a protein multiple sequence alignment that has been computed for a benchmark is only measured with respect to the core columns of the reference alignment. When computing an alignment in practice, however, a reference alignment is not known, so the coreness of its columns can only be predicted. Results: We develop for the first time a predictor of column coreness for protein multiple sequence alignments. This allows us to predict which columns of a computed alignment are core, and hence better estimate the alignment's accuracy. Our approach to predicting coreness is similar to nearest-neighbor classification from machine learning, except we transform nearest-neighbor distances into a coreness prediction via a regression function, and we learn an appropriate distance function through a new optimization formulation that solves a large-scale linear programming problem. We apply our coreness predictor to parameter advising, the task of choosing parameter values for an aligner's scoring function to obtain a more accurate alignment of a specific set of sequences. We show that for this task, our predictor strongly outperforms other column-confidence estimators from the literature, and affords a substantial boost in alignment accuracy.