Improved Sample Complexity for Private Nonsmooth Nonconvex Optimization

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Contribution

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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 $(\epsilon,\delta)$-DP algorithm that returns an $(\alpha,\beta)$-stationary point as long as the dataset is of size $\widetilde{\Omega}(\sqrt{d}/\alpha\beta^{3}+d/\epsilon\alpha\beta^{2})$, 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.