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# Private Stochastic Optimization with Large Worst-Case Lipschitz Parameter: Optimal Rates for (Non-Smooth) Convex Losses and Extension to Non-Convex Losses

*Proceedings of The 34th International Conference on Algorithmic Learning Theory*, PMLR 201:986-1054, 2023.

#### Abstract

We study differentially private (DP) stochastic optimization (SO) with loss functions whose worst-case Lipschitz parameter over all data points may be extremely large. To date, the vast majority of work on DP SO assumes that the loss is uniformly Lipschitz continuous over data (i.e. stochastic gradients are uniformly bounded over all data points). While this assumption is convenient, it often leads to pessimistic excess risk bounds. In many practical problems, the worst-case (uniform) Lipschitz parameter of the loss over all data points may be extremely large due to outliers. In such cases, the error bounds for DP SO, which scale with the worst-case Lipschitz parameter of the loss, are vacuous. To address these limitations, this work provides near-optimal excess risk bounds that do not depend on the uniform Lipschitz parameter of the loss. Building on a recent line of work (Wang et al., 2020; Kamath et al., 2022), we assume that stochastic gradients have bounded $k$-th order \textit{moments} for some $k \geq 2$. Compared with works on uniformly Lipschitz DP SO, our excess risk scales with the $k$-th moment bound instead of the uniform Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers and/or heavy-tailed data. For \textit{convex} and \textit{strongly convex} loss functions, we provide the first asymptotically \textit{optimal} excess risk bounds (up to a logarithmic factor). In contrast to (Wang et al., 2020; Kamath et al., 2022), our bounds do not require the loss function to be differentiable/smooth. We also devise an accelerated algorithm for smooth losses that runs in linear time and has excess risk that is tight in certain practical parameter regimes. Additionally, our work is the first to address \textit{non-convex} non-uniformly Lipschitz loss functions satisfying the \textit{Proximal-PL inequality}; this covers some practical machine learning models. Our Proximal-PL algorithm has near-optimal excess risk.