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We consider stochastic convex optimization for heavy-tailed data with theguarantee of being differentially private (DP). Prior work on this problem isrestricted to the gradient descent (GD) method, which is inefficient forlarge-scale problems. In this paper, we resolve this issue and derive the firsthigh-probability bounds for the private stochastic method with clipping. Forgeneral convex problems, we derive excess population risks$\Tilde{O}\left(\frac{d^{1/7}\sqrt{\ln\frac{(n \epsilon)^2}{\betad}}}{(n\epsilon)^{2/7}}\right)$ and$\Tilde{O}\left(\frac{d^{1/7}\ln\frac{(n\epsilon)^2}{\betad}}{(n\epsilon)^{2/7}}\right)$ under bounded or unbounded domain assumption,respectively (here $n$ is the sample size, $d$ is the dimension of the data,$\beta$ is the confidence level and $\epsilon$ is the private level). Then, weextend our analysis to the strongly convex case and non-smooth case (whichworks for generalized smooth objectives with H$\ddot{\text{o}}$lder-continuousgradients). We establish new excess risk bounds without bounded domainassumption. The results above achieve lower excess risks and gradientcomplexities than existing methods in their corresponding cases. Numericalexperiments are conducted to justify the theoretical improvement.

Efficient Private SCO for Heavy-Tailed Data via Clipping