A Stochastic Sampling Approach to Privacy

This paper proposes an optimal stochastic sampling approach to privacy, in which a sensor observes a process which is correlated to private information. In out set-up, a sampler decides to keep or discard the sensor's observations. The kept samples are shared with an adversary who might attempt to infer the private process based on the sampler's output. The privacy leakages are captured with the mutual information between the private process and sampler's output. We cast the optimal sampling design as an optimization problem with two objectives: (i) minimizing the reconstruction error of the observed process using the sampler's output, (ii) reducing the privacy leakages. We first show the optimal reconstruction policy is deterministic and can be obtained by solving a one-step optimization problem at each time step. We also derive the optimality equations of the privacy-sampler for a general class of processes via the dynamic decomposition method, and show the sampler controls the adversary's belief about the private input. Also, we propose a simplified design for linear Gaussian processes by restricting the sampling policy to a special collection. We show that the optimal reconstruction of the system state and the private process is similar to Kalman filter in the linear Gaussian case, and the objective of the sampler design problem can be analytically expressed based on a conditional mean and covariance matrix. Furthermore, we develop an numerical algorithm to optimize the sampling and reconstruction policies, wherein the policy gradient theorem for the optimal sampling design is derived based on the implicit function theorem. Finally, we verify our design and show it capabilities in state reconstruction, privacy protection and data size reduction via simulations.