Privacy Risk for anisotropic Langevin dynamics using relative entropy bounds

The privacy preserving properties of Langevin dynamics with additive isotropic noise have been extensively studied. However, the isotropic noise assumption is very restrictive: (a) when adding noise to existing learning algorithms to preserve privacy and maintain the best possible accuracy one should take into account the relative magnitude of the outputs and their correlations; (b) popular algorithms such as stochastic gradient descent (and their continuous time limits) appear to possess anisotropic covariance properties. To study the privacy risks for the anisotropic noise case, one requires general results on the relative entropy between the laws of two Stochastic Differential Equations with different drifts and diffusion coefficients. Our main contribution is to establish such a bound using stability estimates for solutions to the Fokker-Planck equations via functional inequalities. With additional assumptions, the relative entropy bound implies an $(\epsilon,\delta)$-differential privacy bound. We discuss the practical implications of our bound related to privacy risk in different contexts.Finally, the benefits of anisotropic noise are illustrated using numerical results on optimising a quadratic loss or calibrating a neural network.