We present a differentially private extension of the block coordinate descent based on objective perturbation. The algorithm iteratively performs linear regression in a federated setting on vertically partitioned data. In addition to a privacy guarantee, the algorithm also offers a utility guarantee; a tolerance parameter indicates how much the differentially private regression may deviate from an analysis without differential privacy. The algorithm’s performance is compared with the standard block coordinate descent algorithm and the trade-off between utility and privacy is studied. The performance is studied using both artificial test data and the forest fires data set. We find that the algorithm is fast and able to generate practical predictions with single-digit privacy budgets, albeit with some accuracy loss.

Phylogenomic analyses have revolutionized the study of biodiversity, but they have revealed that estimated tree topologies can depend, at least in part, on the subset of the genome that is analyzed. For example, estimates of trees for avian orders differ if protein coding or non-coding data are analyzed. The bird tree is a good study system because the historical signal for relationships among orders is very weak, which should permit subtle non-historical signals to be identified, while monophyly of orders is strongly corroborated, allowing identification of strong non-historical signals. Hydrophobic amino acids in mitochondrially-encoded proteins, which are expected to be found in transmembrane helices, have been hypothesized to be associated with non-historical signals. We tested this hypothesis by comparing the evolution of transmembrane helices and extramembrane segments of mitochondrial proteins from 420 bird species, sampled from most avian orders. We estimated amino acids exchangeabilities for both structural environments and assessed the performance of phylogenetic analysis using each data type. We compared those relative exchangeabilities with values calculated using a substitution dataset for transmembrane helices from a variety of sampled set of nuclear- and mitochondrially-encoded proteins, allowing us to compare the bird-specific mitochondrial models with a general model of transmembrane protein evolution. To complement our amino acid analyses, we examined the impact of protein structure on patterns of nucleotide evolution. Models of transmembrane and extramembrane sequence evolution for amino acids and nucleotides exhibited striking differences, but there was no evidence for strong topological data type effects. However, incorporating protein structure into analyses of mitochondrially-encoded proteins improved model fit. Thus, we believe that considering protein structure will improve analyses of mitogenomic data, both in birds and in other taxa.