  In many applications, multiple parties have private data regarding the same
set of users but on disjoint sets of attributes, and a server wants to leverage
the data to train a model. To enable model learning while protecting the
privacy of the data subjects, we need vertical federated learning (VFL)
techniques, where the data parties share only information for training the
model, instead of the private data. However, it is challenging to ensure that
the shared information maintains privacy while learning accurate models. To the
best of our knowledge, the algorithm proposed in this paper is the first
practical solution for differentially private vertical federated k-means
clustering, where the server can obtain a set of global centers with a provable
differential privacy guarantee. Our algorithm assumes an untrusted central
server that aggregates differentially private local centers and membership
encodings from local data parties. It builds a weighted grid as the synopsis of
the global dataset based on the received information. Final centers are
generated by running any k-means algorithm on the weighted grid. Our approach
for grid weight estimation uses a novel, light-weight, and differentially
private set intersection cardinality estimation algorithm based on the
Flajolet-Martin sketch. To improve the estimation accuracy in the setting with
more than two data parties, we further propose a refined version of the weights
estimation algorithm and a parameter tuning strategy to reduce the final
k-means utility to be close to that in the central private setting. We provide
theoretical utility analysis and experimental evaluation results for the
cluster centers computed by our algorithm and show that our approach performs
better both theoretically and empirically than the two baselines based on
existing techniques.
