Daogao Liu

Sivakanth Gopi, Yin Tat Lee, Daogao Liu

In this paper, we study private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$. We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling from $π(x)\propto \exp(-k(F(x)+μ\|x\|_2^2/2))$ recovers both the known optimal empirical risk and population loss under $(ε,δ)$-DP. Furthermore, we show how to implement this mechanism using $\widetilde{O}(n \min(d, n))$ queries to $f_i(x)$ for the DP-SCO where $n$ is the number of samples/users and $d$ is the ambient dimension. We also give a (nearly) matching lower bound $\widetildeΩ(n \min(d, n))$ on the number of evaluation queries. Our results utilize the following tools that are of independent interest: (1) We prove Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz. Our privacy bound is \emph{optimal} as it includes the privacy of Gaussian mechanism as a special case and is proved using the isoperimetric inequality for strongly log-concave measures. (2) We show how to sample from $\exp(-F(x)-μ\|x\|^2_2/2)$ for $G$-Lipschitz $F$ with $η$ error in total variation (TV) distance using $\widetilde{O}((G^2/μ) \log^2(d/η))$ unbiased queries to $F(x)$. This is the first sampler whose query complexity has \emph{polylogarithmic dependence} on both dimension $d$ and accuracy $η$.

Sivakanth Gopi, Yin Tat Lee, Daogao Liu et al.

We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm $\|\cdot\|$. Our algorithms are based on a regularized exponential mechanism which samples from the density $\propto \exp(-k(F+μr))$ where $F$ is the empirical loss and $r$ is a regularizer which is strongly convex with respect to $\|\cdot\|$, generalizing a recent work of [Gopi, Lee, Liu '22] 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 $ε\to \infty$. 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 [Asi, Feldman, Koren, Talwar '21, Bassily, Guzman, Nandi '21] 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.

Sivakanth Gopi, Yin Tat Lee, Daogao Liu et al.

The development of efficient sampling algorithms catering to non-Euclidean geometries has been a challenging endeavor, as discretization techniques which succeed in the Euclidean setting do not readily carry over to more general settings. We develop a non-Euclidean analog of the recent proximal sampler of [LST21], which naturally induces regularization by an object known as the log-Laplace transform (LLT) of a density. We prove new mathematical properties (with an algorithmic flavor) of the LLT, such as strong convexity-smoothness duality and an isoperimetric inequality, which are used to prove a mixing time on our proximal sampler matching [LST21] under a warm start. As our main application, we show our warm-started sampler improves the value oracle complexity of differentially private convex optimization in $\ell_p$ and Schatten-$p$ norms for $p \in [1, 2]$ to match the Euclidean setting [GLL22], while retaining state-of-the-art excess risk bounds [GLLST23]. We find our investigation of the LLT to be a promising proof-of-concept of its utility as a tool for designing samplers, and outline directions for future exploration.