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The least squares problem with L1-regularized regressors, called Lasso, is awidely used approach in optimization problems where sparsity of the regressorsis desired. This formulation is fundamental for many applications in signalprocessing, machine learning and control. As a motivating problem, weinvestigate a sparse data predictive control problem, run at a cloud service tocontrol a system with unknown model, using L1-regularization to limit thebehavior complexity. The input-output data collected for the system isprivacy-sensitive, hence, we design a privacy-preserving solution usinghomomorphically encrypted data. The main challenges are the non-smoothness ofthe L1-norm, which is difficult to evaluate on encrypted data, as well as theiterative nature of the Lasso problem. We use a distributed ADMM formulationthat enables us to exchange substantial local computation for littlecommunication between multiple servers. We first give an encrypted multi-partyprotocol for solving the distributed Lasso problem, by approximating thenon-smooth part with a Chebyshev polynomial, evaluating it on encrypted data,and using a more cost effective distributed bootstrapping operation. For theexample of data predictive control, we prefer a non-homogeneous splitting ofthe data for better convergence. We give an encrypted multi-party protocol forthis non-homogeneous splitting of the Lasso problem to a non-homogeneous set ofservers: one powerful server and a few less powerful devices, added forsecurity reasons. Finally, we provide numerical results for our proposedsolutions.

Encrypted Distributed Lasso for Sparse Data Predictive Control