We develop algorithms for private stochastic convex optimization that adapt to the hardness of the specific function we wish to optimize. While previous work provide worst-case bounds for arbitrary convex functions, it is often the case that the function at hand belongs to a smaller class that enjoys faster rates. Concretely, we show that for functions exhibiting $\kappa$-growth around the optimum, i.e., $f(x) \ge f(x^*) + \lambda \kappa^{-1} \|x-x^*\|_2^\kappa$ for $\kappa > 1$, our algorithms improve upon the standard ${\sqrt{d}}/{n\varepsilon}$ privacy rate to the faster $({\sqrt{d}}/{n\varepsilon})^{\tfrac{\kappa}{\kappa - 1}}$. Crucially, they achieve these rates without knowledge of the growth constant $\kappa$ of the function. Our algorithms build upon the inverse sensitivity mechanism, which adapts to instance difficulty (Asi & Duchi, 2020), and recent localization techniques in private optimization (Feldman et al., 2020). We complement our algorithms with matching lower bounds for these function classes and demonstrate that our adaptive algorithm is \emph{simultaneously} (minimax) optimal over all $\kappa \ge 1+c$ whenever $c = \Theta(1)$.

翻译：我们开发了符合我们想要优化的具体功能的硬度的私人软盘配置优化算法。 虽然先前的工作为任意 convex 函数提供了最坏的框框, 但通常情况下, 我们手头的函数属于一个享有更快率的较小等级。 具体地说, 我们显示, 对于显示美元( kappa) 在最佳值周围增长的功能, 即 $( x)\ gf( x) +\ lumbda\ kappa)-1}, $( kappa) > 1美元, 提供最坏的框框框框框。 虽然先前的工作为 $( kapppa) 提供了最差的框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框框