Mirror Descent Algorithms with Nearly Dimension-Independent Rates for Differentially-Private Stochastic Saddle-Point Problems

We study the problem of differentially-private (DP) stochastic (convex-concave) saddle-points in the $\ell_1$ setting. We propose $(\varepsilon, \delta)$-DP algorithms based on stochastic mirror descent that attain nearly dimension-independent convergence rates for the expected duality gap, a type of guarantee that was known before only for bilinear objectives. For convex-concave and first-order-smooth stochastic objectives, our algorithms attain a rate of $\sqrt{\log(d)/n} + (\log(d)^{3/2}/[n\varepsilon])^{1/3}$, where $d$ is the dimension of the problem and $n$ the dataset size. Under an additional second-order-smoothness assumption, we show that the duality gap is bounded by $\sqrt{\log(d)/n} + \log(d)/\sqrt{n\varepsilon}$ with high probability, by using bias-reduced gradient estimators. This rate provides evidence of the near-optimality of our approach, since a lower bound of $\sqrt{\log(d)/n} + \log(d)^{3/4}/\sqrt{n\varepsilon}$ exists. Finally, we show that combining our methods with acceleration techniques from online learning leads to the first algorithm for DP Stochastic Convex Optimization in the $\ell_1$ setting that is not based on Frank-Wolfe methods. For convex and first-order-smooth stochastic objectives, our algorithms attain an excess risk of $\sqrt{\log(d)/n} + \log(d)^{7/10}/[n\varepsilon]^{2/5}$, and when additionally assuming second-order-smoothness, we improve the rate to $\sqrt{\log(d)/n} + \log(d)/\sqrt{n\varepsilon}$. Instrumental to all of these results are various extensions of the classical Maurey Sparsification Lemma \cite{Pisier:1980}, which may be of independent interest.