On Differentially Private Stochastic Convex Optimization with Heavy-tailed Data

• Published in: arXiv.org 2020-10
• Main Authors: Wang, Di, Xiao, Hanshen, Devadas, Srini, Xu, Jinhui
• Format: Article
• Language: English
• Subjects: Algorithms; Computational geometry; Convex analysis; Convexity; Irregularities; Optimization
• ISSN: 2331-8422

In this paper, we consider the problem of designing Differentially Private (DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data. The irregularity of such data violates some key assumptions used in almost all existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP guarantees. To better understand this type of challenges, we provide in this paper a comprehensive study of DP-SCO under various settings. First, we consider the case where the loss function is strongly convex and smooth. For this case, we propose a method based on the sample-and-aggregate framework, which has an excess population risk of \(\tilde{O}(\frac{d^3}{n\epsilon^4})\) (after omitting other factors), where \(n\) is the sample size and \(d\) is the dimensionality of the data. Then, we show that with some additional assumptions on the loss functions, it is possible to reduce the \textit{expected} excess population risk to \(\tilde{O}(\frac{ d^2}{ n\epsilon^2 })\). To lift these additional conditions, we also provide a gradient smoothing and trimming based scheme to achieve excess population risks of \(\tilde{O}(\frac{ d^2}{n\epsilon^2})\) and \(\tilde{O}(\frac{d^\frac{2}{3}}{(n\epsilon^2)^\frac{1}{3}})\) for strongly convex and general convex loss functions, respectively, \textit{with high probability}. Experiments suggest that our algorithms can effectively deal with the challenges caused by data irregularity.