## Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data

### Gautam Kamath · Xingtu Liu · Huanyu Zhang

##### Hall E #1006

Keywords: [ OPT: Convex ] [ OPT: Stochastic ] [ T: Learning Theory ] [ T: Optimization ] [ SA: Privacy-preserving Statistics and Machine Learning ]

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Wed 20 Jul 3:30 p.m. PDT — 5:30 p.m. PDT

Oral presentation: SA: Trustworthy Machine Learning
Wed 20 Jul 10:15 a.m. PDT — 11:45 a.m. PDT

Abstract: We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy (DP). Most prior work on this problem is restricted to the case where the loss function is Lipschitz. Instead, as introduced by Wang, Xiao, Devadas, and Xu~\cite{WangXDX20}, we study general convex loss functions with the assumption that the distribution of gradients has bounded $k$-th moments. We provide improved upper bounds on the excess population risk under concentrated DP for convex and strongly convex loss functions. Along the way, we derive new algorithms for private mean estimation of heavy-tailed distributions, under both pure and concentrated DP. Finally, we prove nearly-matching lower bounds for private stochastic convex optimization with strongly convex losses and mean estimation, showing new separations between pure and concentrated DP.

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