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Consider two data providers, each maintaining private records of differentfeature sets about common entities. They aim to learn a linear model jointly ina federated setting, namely, data is local and a shared model is trained fromlocally computed updates. In contrast with most work on distributed learning,in this scenario (i) data is split vertically, i.e. by features, (ii) only onedata provider knows the target variable and (iii) entities are not linkedacross the data providers. Hence, to the challenge of private learning, we addthe potentially negative consequences of mistakes in entity resolution. Ourcontribution is twofold. First, we describe a three-party end-to-end solutionin two phases ---privacy-preserving entity resolution and federated logisticregression over messages encrypted with an additively homomorphic scheme---,secure against a honest-but-curious adversary. The system allows learningwithout either exposing data in the clear or sharing which entities the dataproviders have in common. Our implementation is as accurate as a naivenon-private solution that brings all data in one place, and scales to problemswith millions of entities with hundreds of features. Second, we provide what isto our knowledge the first formal analysis of the impact of entity resolution'smistakes on learning, with results on how optimal classifiers, empiricallosses, margins and generalisation abilities are affected. Our results bring aclear and strong support for federated learning: under reasonable assumptionson the number and magnitude of entity resolution's mistakes, it can beextremely beneficial to carry out federated learning in the setting where eachpeer's data provides a significant uplift to the other.

Private federated learning on vertically partitioned data via entity resolution and additively homomorphic encryption