Private Convex Optimization in General Norms

We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $\normx{\cdot}$. Our algorithms are based on a regularized exponential mechanism which samples from the density $\propto \exp(-k(F+\mu r))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $\normx{\cdot}$, generalizing a recent work of \cite{GLL22} to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization), by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces, and directly recovers non-private SCO rates achieved by mirror descent, as the privacy parameter $\eps \to \infty$. As applications, for Lipschitz optimization in $\ell_p$ norms for all $p \in (1, 2)$, we obtain the first optimal privacy-utility tradeoffs; for $p = 1$, we improve tradeoffs obtained by the recent works \cite{AsiFKT21, BassilyGN21} by at least a logarithmic factor. Our $\ell_p$ norm and Schatten-$p$ norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.