  The least squares problem with L1-regularized regressors, called Lasso, is a
widely used approach in optimization problems where sparsity of the regressors
is desired. This formulation is fundamental for many applications in signal
processing, machine learning and control. As a motivating problem, we
investigate a sparse data predictive control problem, run at a cloud service to
control a system with unknown model, using L1-regularization to limit the
behavior complexity. The input-output data collected for the system is
privacy-sensitive, hence, we design a privacy-preserving solution using
homomorphically encrypted data. The main challenges are the non-smoothness of
the L1-norm, which is difficult to evaluate on encrypted data, as well as the
iterative nature of the Lasso problem. We use a distributed ADMM formulation
that enables us to exchange substantial local computation for little
communication between multiple servers. We first give an encrypted multi-party
protocol for solving the distributed Lasso problem, by approximating the
non-smooth part with a Chebyshev polynomial, evaluating it on encrypted data,
and using a more cost effective distributed bootstrapping operation. For the
example of data predictive control, we prefer a non-homogeneous splitting of
the data for better convergence. We give an encrypted multi-party protocol for
this non-homogeneous splitting of the Lasso problem to a non-homogeneous set of
servers: one powerful server and a few less powerful devices, added for
security reasons. Finally, we provide numerical results for our proposed
solutions.
