Adapting to function difficulty and growth conditions in private optimization

Hilal Asi, Daniel Asher Nathan Levy, John Duchi

Keywords: privacy, differential privacy, adaptivity, optimization, convex, learning, sco, lower bounds, minimax, growth

Abstract: 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., $f(x) \ge f(x^\star) + \lambda \kappa^{-1} \|x-x^\star\|_2^\kappa$ 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 [2], and recent localization techniques in private optimization [25]. We complement our algorithms with matching lower bounds for these function classes and demonstrate that our adaptive algorithm is simultaneously (minimax) optimal over all $\kappa \ge 1+c$ whenever $c = \Theta(1)$.

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TL;DR: We provide private adaptive algorithms that adapt to the growth of the actual instance and improves upon standard worst-case rates and show that on easier functions, the extra error incurred by privacy shrinks significantly.