Free access
Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

Private Convex Optimization in General Norms


We propose a new framework for differentially private optimization of convex functions which are Lipschitz in an arbitrary norm ||·||x. Our algorithms are based on a regularized exponential mechanism which samples from the density ∞ exp(-k(F + μr)) where F is the empirical loss and τ is a regularizer which is strongly convex with respect to ||·||x, generalizing a recent work of [GLL22] to non-Euclidean settings. We show that this mechanism satisfies Gaussian differential privacy and solves both DP-ERM (empirical risk minimization) and DP-SCO (stochastic convex optimization), by using localization tools from convex geometry. Our framework is the first to apply to private convex optimization in general normed spaces, and directly recovers non-private SCO rates achieved by mirror descent, as the privacy parameter ε → ∞. As applications, for Lipschitz optimization in ℓp norms for all p ∈ (1, 2), we obtain the first optimal privacy-utility tradeoffs; for p = 1, we improve tradeoffs obtained by the recent works [AFKT21, BGN21] by at least a logarithmic factor. Our ℓp norm and Schatten-p norm optimization frameworks are complemented with polynomial-time samplers whose query complexity we explicitly bound.

Formats available

You can view the full content in the following formats:

Information & Authors


Published In

cover image Proceedings
Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 5068 - 5089
Editors: Nikhil Bansal, University of Michigan, Ann Arbor, Michigan, USA and Viswanath Nagarajan, University of Michigan, Ann Arbor, Michigan, USA
ISBN (Online): 978-1-61197-755-4


Published online: 16 January 2023



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options


View PDF

Get Access







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account