Private Stochastic Optimization in the Presence of Outliers: Optimal Rates for (Non-Smooth) Convex Losses and Extension to Non-Convex Losses

We study differentially private (DP) stochastic optimization (SO) with data containing outliers and loss functions that are (possibly) not Lipschitz continuous. To date, the vast majority of work on DP SO assumes that the loss is uniformly Lipschitz over data (i.e. stochastic gradients are uniformly bounded over all data points). While this assumption is convenient, it is often unrealistic: in many practical problems, the loss function may not be uniformly Lipschitz. Even when the loss function is Lipschitz continuous, the worst-case 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 does not require the loss function to be uniformly Lipschitz. Instead, building on a recent line of work [WXDX20, KLZ22], we make the weaker assumption that stochastic gradients have bounded $k$-th order moments for some $k \geq 2$. Compared with works on DP Lipschitz SO, our excess risk scales with the $k$-th moment bound instead of the Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers. For convex and strongly convex loss functions, we provide the first asymptotically optimal excess risk bounds (up to a logarithmic factor). In contrast to the prior works, 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 non-convex non-Lipschitz loss functions satisfying the Proximal-PL inequality; this covers some practical machine learning models. Our Proximal-PL algorithm has near-optimal excess risk.