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arXiv · 2410.05880

Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization

Guy Kornowski·Daogao Liu·Kunal Talwar
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Abstract

We study differentially private (DP) optimization algorithms for stochastic and empirical objectives which are neither smooth nor convex, and propose methods that return a Goldstein-stationary point with sample complexity bounds that improve on existing works. We start by providing a single-pass $(ε,δ)$-DP algorithm that returns an $(α,β)$-stationary point as long as the dataset is of size $\widetildeΩ(\sqrt{d}/αβ^{3}+d/εαβ^{2})$, which is $Ω(\sqrt{d})$ times smaller than the algorithm of Zhang et al. [2024] for this task, where $d$ is the dimension. We then provide a multi-pass polynomial time algorithm which further improves the sample complexity to $\widetildeΩ\left(d/β^2+d^{3/4}/εα^{1/2}β^{3/2}\right)$, by designing a sample efficient ERM algorithm, and proving that Goldstein-stationary points generalize from the empirical loss to the population loss.

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Guy Kornowski, Daogao Liu, Kunal Talwar. 2025-06-07. Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization. https://arxiv.org/abs/2410.05880

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