Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization

• URL: http://arxiv.org/abs/2410.05880v1
• Date: Tue, 8 Oct 2024 10:15:49 GMT
• Title: Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization
• Authors: Guy Kornowski, Daogao Liu, Kunal Talwar,
• Abstract summary: We study differentially private (DP) optimization algorithms for and empirical objectives which are neither smooth nor convex. We provide a single-pass $(alpha,beta)$-DP algorithm that returns an $widetildeOmegaleft (1/alphabeta3+d/epsilonalphabeta2+d3/4/epsilonalpha1/2beta3/2right)$. We then provide a multi-pass time algorithm which further improves the sample complexity to $widetildeOmegaleft(
• Score: 28.497079108813924
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $(\epsilon,\delta)$-DP algorithm that returns an $(\alpha,\beta)$-stationary point as long as the dataset is of size $\widetilde{\Omega}\left(1/\alpha\beta^{3}+d/\epsilon\alpha\beta^{2}+d^{3/4}/\epsilon^{1/2}\alpha\beta^{5/2}\right)$, which is $\Omega(\sqrt{d})$ 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 $\widetilde{\Omega}\left(d/\beta^2+d^{3/4}/\epsilon\alpha^{1/2}\beta^{3/2}\right)$, 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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