  We consider a fully-decentralized scenario in which no central trusted entity
exists and all clients are honest-but-curious. The state-of-the-art approaches
to this problem often rely on cryptographic protocols, such as multiparty
computation (MPC), that require mapping real-valued data to a discrete
alphabet, specifically a finite field. These approaches, however, can result in
substantial accuracy losses due to computation overflows. To address this
issue, we propose A-MPC, a private analog MPC protocol that performs all
computations in the analog domain. We characterize the privacy of individual
datasets in terms of $(\epsilon, \delta)$-local differential privacy, where the
privacy of a single record in each client's dataset is guaranteed against other
participants. In particular, we characterize the required noise variance in the
Gaussian mechanism in terms of the required $(\epsilon,\delta)$-local
differential privacy parameters by solving an optimization problem.
Furthermore, compared with existing decentralized protocols, A-MPC keeps the
privacy of individual datasets against the collusion of all other participants,
thereby, in a notably significant improvement, increasing the maximum number of
colluding clients tolerated in the protocol by a factor of three compared with
the state-of-the-art collaborative learning protocols. Our experiments
illustrate that the accuracy of the proposed $(\epsilon,\delta)$-locally
differential private logistic regression and linear regression models trained
in a fully-decentralized fashion using A-MPC closely follows that of a
centralized one performed by a single trusted entity.
