Differentially Private Stochastic Optimization: New Results in Convex and Non-Convex Settings

Abstract: We study differentially private stochastic optimization in convex and non-convex settings. For the convex case, we focus on the family of non-smooth generalized linear losses (GLLs). Our algorithm for the $\ell_2$ setting achieves optimal excess population risk in near-linear time, while the best known differentially private algorithms for general convex losses run in super-linear time. Our algorithm for the $\ell_1$ setting has nearly-optimal excess population risk $$\tilde{O}\big(\sqrt{\frac{\log{d}}{n\varepsilon}}\big)$$, and circumvents the dimension dependent lower bound of \cite{Asi:2021} for general non-smooth convex losses. In the differentially private non-convex setting, we provide several new algorithms for approximating stationary points of the population risk. For the $\ell_1$-case with smooth losses and polyhedral constraint, we provide the first nearly dimension independent rate, $$\tilde O\big(\frac{\log<sup>{2/3}{d}}{{(n\varepsilon)<sup>{1/3}}}\big)$$ in linear time. For the constrained $\ell_2$-case with smooth losses, we obtain a linear-time algorithm with rate $$\tilde O\big(\frac{1}{n<sup>{1/3}}+\frac{d<sup>{1/5}}{(n\varepsilon)<sup>{2/5}}\big)$$. Finally, for the $\ell_2$-case we provide the first method for {\em non-smooth weakly convex} stochastic optimization with rate $$\tilde O\big(\frac{1}{n<sup>{1/4}}+\frac{d<sup>{1/6}}{(n\varepsilon)<sup>{1/3}}\big)$$ which matches the best existing non-private algorithm when $d= O(\sqrt{n})$. We also extend all our results above for the non-convex $\ell_2$ setting to the $\ell_p$ setting, where $1 &lt; p \leq 2$, with only polylogarithmic (in the dimension) overhead in the rates.