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We develop algorithms for private stochastic convex optimization that adaptto the hardness of the specific function we wish to optimize. While previouswork provide worst-case bounds for arbitrary convex functions, it is often thecase that the function at hand belongs to a smaller class that enjoys fasterrates. Concretely, we show that for functions exhibiting $\kappa$-growth aroundthe 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, theyachieve these rates without knowledge of the growth constant $\kappa$ of thefunction. Our algorithms build upon the inverse sensitivity mechanism, whichadapts to instance difficulty (Asi & Duchi, 2020), and recent localizationtechniques in private optimization (Feldman et al., 2020). We complement ouralgorithms with matching lower bounds for these function classes anddemonstrate that our adaptive algorithm is \emph{simultaneously} (minimax)optimal over all $\kappa \ge 1+c$ whenever $c = \Theta(1)$.

Adapting to Function Difficulty and Growth Conditions in Private Optimization