Faster Differentially Private Samplers via Rényi Divergence Analysis of Discretized Langevin MCMC
This work addresses a bottleneck in implementing differential privacy efficiently for high-dimensional data, offering a provably faster alternative to existing methods.
The paper tackles the computational challenge of sampling from high-dimensional distributions for differentially private algorithms by establishing rapid convergence of Langevin dynamics-based methods under Rényi divergence, leading to faster private algorithms for smooth, strongly-convex functions.
Various differentially private algorithms instantiate the exponential mechanism, and require sampling from the distribution $\exp(-f)$ for a suitable function $f$. When the domain of the distribution is high-dimensional, this sampling can be computationally challenging. Using heuristic sampling schemes such as Gibbs sampling does not necessarily lead to provable privacy. When $f$ is convex, techniques from log-concave sampling lead to polynomial-time algorithms, albeit with large polynomials. Langevin dynamics-based algorithms offer much faster alternatives under some distance measures such as statistical distance. In this work, we establish rapid convergence for these algorithms under distance measures more suitable for differential privacy. For smooth, strongly-convex $f$, we give the first results proving convergence in Rényi divergence. This gives us fast differentially private algorithms for such $f$. Our techniques and simple and generic and apply also to underdamped Langevin dynamics.