Langevin Diffusion: An Almost Universal Algorithm for Private Euclidean (Convex) Optimization

In this paper we revisit the problem of differentially private empirical risk minimization (DP-ERM) and differentially private stochastic convex optimization (DP-SCO). We show that a well-studied continuous time algorithm from statistical physics, called Langevin diffusion (LD), simultaneously provides optimal privacy/utility trade-offs for both DP-ERM and DP-SCO, under $\epsilon$-DP, and $(\epsilon,\delta)$-DP. Using the uniform stability properties of LD, we provide the optimal excess population risk guarantee for $\ell_2$-Lipschitz convex losses under $\epsilon$-DP, which was an open problem since [BST14]. Along the way, we provide various technical tools, which can be of independent interest: i) A new R\'enyi divergence bound for LD, when run on loss functions over two neighboring data sets, ii) Excess empirical risk bounds for last-iterate LD, analogous to that of Shamir and Zhang for noisy stochastic gradient descent (SGD), and iii) A two phase excess risk analysis of LD, where the first phase is when the diffusion has not converged in any reasonable sense to a stationary distribution, and in the second phase when the diffusion has converged to a variant of Gibbs distribution. Our universality results crucially rely on the dynamics of LD. When it has converged to a stationary distribution, we obtain the optimal bounds under $\epsilon$-DP. When it is run only for a very short time $\propto 1/p$, we obtain the optimal bounds under $(\epsilon,\delta)$-DP. Here, $p$ is the dimensionality of the model space. Our work initiates a systematic study of DP continuous time optimization. We believe this may have ramifications in the design of discrete time DP optimization algorithms analogous to that in the non-private setting, where continuous time dynamical viewpoints have helped in designing new algorithms, including the celebrated mirror-descent and Polyak's momentum method.