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Differential private optimization for nonconvex smooth objective isconsidered. In the previous work, the best known utility bound is $\widetildeO(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ in terms of the squared full gradientnorm, which is achieved by Differential Private Gradient Descent (DP-GD) as aninstance, where $n$ is the sample size, $d$ is the problem dimensionality and$\varepsilon_\mathrm{DP}$ is the differential privacy parameter. To improve thebest known utility bound, we propose a new differential private optimizationframework called \emph{DIFF2 (DIFFerential private optimization via gradientDIFFerences)} that constructs a differential private global gradient estimatorwith possibly quite small variance based on communicated \emph{gradientdifferences} rather than gradients themselves. It is shown that DIFF2 with agradient descent subroutine achieves the utility of $\widetildeO(d^{2/3}/(n\varepsilon_\mathrm{DP})^{4/3})$, which can be significantly betterthan the previous one in terms of the dependence on the sample size $n$. To thebest of our knowledge, this is the first fundamental result to improve thestandard utility $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ fornonconvex objectives. Additionally, a more computational and communicationefficient subroutine is combined with DIFF2 and its theoretical analysis isalso given. Numerical experiments are conducted to validate the superiority ofDIFF2 framework.

DIFF2: Differential Private Optimization via Gradient Differences for Nonconvex Distributed Learning