Non-Euclidean Differentially Private Stochastic Convex Optimization

• URL: http://arxiv.org/abs/2103.01278v1
• Date: Mon, 1 Mar 2021 19:48:44 GMT
• Title: Non-Euclidean Differentially Private Stochastic Convex Optimization
• Authors: Raef Bassily, Crist\'obal Guzm\'an, Anupama Nandi
• Abstract summary: We show that noisy gradient descent (SGD) algorithms attain the optimal excess risk in low-dimensional regimes. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.
• Score: 15.302167005107135
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a constraint set with bounded $\ell_2$ diameter. Algorithms based on noisy stochastic gradient descent (SGD) are known to attain the optimal excess risk in this setting. In this work, we conduct a systematic study of DP-SCO for $\ell_p$-setups. For $p=1$, under a standard smoothness assumption, we give a new algorithm with nearly optimal excess risk. This result also extends to general polyhedral norms and feasible sets. For $p\in(1, 2)$, we give two new algorithms, whose central building block is a novel privacy mechanism, which generalizes the Gaussian mechanism. Moreover, we establish a lower bound on the excess risk for this range of $p$, showing a necessary dependence on $\sqrt{d}$, where $d$ is the dimension of the space. Our lower bound implies a sudden transition of the excess risk at $p=1$, where the dependence on $d$ changes from logarithmic to polynomial, resolving an open question in prior work [TTZ15] . For $p\in (2, \infty)$, noisy SGD attains optimal excess risk in the low-dimensional regime; in particular, this proves the optimality of noisy SGD for $p=\infty$. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.

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