  Consider two data providers, each maintaining private records of different
feature sets about common entities. They aim to learn a linear model jointly in
a federated setting, namely, data is local and a shared model is trained from
locally computed updates. In contrast with most work on distributed learning,
in this scenario (i) data is split vertically, i.e. by features, (ii) only one
data provider knows the target variable and (iii) entities are not linked
across the data providers. Hence, to the challenge of private learning, we add
the potentially negative consequences of mistakes in entity resolution. Our
contribution is twofold. First, we describe a three-party end-to-end solution
in two phases ---privacy-preserving entity resolution and federated logistic
regression over messages encrypted with an additively homomorphic scheme---,
secure against a honest-but-curious adversary. The system allows learning
without either exposing data in the clear or sharing which entities the data
providers have in common. Our implementation is as accurate as a naive
non-private solution that brings all data in one place, and scales to problems
with millions of entities with hundreds of features. Second, we provide what is
to our knowledge the first formal analysis of the impact of entity resolution's
mistakes on learning, with results on how optimal classifiers, empirical
losses, margins and generalisation abilities are affected. Our results bring a
clear and strong support for federated learning: under reasonable assumptions
on the number and magnitude of entity resolution's mistakes, it can be
extremely beneficial to carry out federated learning in the setting where each
peer's data provides a significant uplift to the other.
