We study the problem of differentially private stochastic convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a k {\text{th}} -moment bound on the Lipschitz constants of sample functions, rather than a uniform bound. We propose a new reduction-based approach that enables us to obtain the first optimal rates (up to logarithmic factors) in the heavy-tailed setting, achieving error G_2 \cdot \frac 1 {\sqrt n} G_k \cdot (\frac{\sqrt d}{n\epsilon}) {1 - \frac 1 k} under (\epsilon, \delta) -approximate differential privacy, up to a mild \textup{polylog}(\frac{1}{\delta}) factor, where G_2 2 and G_k k are the 2 {\text{nd}} and k {\text{th}} moment bounds on sample Lipschitz constants, nearly-matching a lower bound of [LR23].We then give a suite of private algorithms for DP-SCO with heavy-tailed gradients improving our basic result under additional assumptions, including an optimal algorithm under a known-Lipschitz constant assumption, a near-linear time algorithm for smooth functions, and an optimal linear time algorithm for smooth generalized linear models.

User-level differentially private stochastic convex optimization (DP-SCO) has garnered significant attention due to the paramount importance of safeguarding user privacy in modern large-scale machine learning applications. Current methods, such as those based on differentially private stochastic gradient descent (DP-SGD), often struggle with high noise accumulation and suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a novel linear-time algorithm that leverages robust statistics, specifically the median and trimmed mean, to overcome these challenges. Our approach uniquely bounds the sensitivity of all intermediate iterates of SGD with gradient estimation based on robust statistics, thereby significantly reducing the gradient estimation noise for privacy purposes and enhancing the privacy-utility trade-off. By sidestepping the repeated privatization required by previous methods, our algorithm not only achieves an improved theoretical privacy-utility trade-off but also maintains computational efficiency. We complement our algorithm with an information-theoretic lower bound, showing that our upper bound is optimal up to logarithmic factors and the dependence on $\epsilon$. This work sets the stage for more robust and efficient privacy-preserving techniques in machine learning, with implications for future research and application in the field.

We study the problem of differentially private stochastic convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a $k^{\text{th}}$-moment bound on the Lipschitz constants of sample functions rather than a uniform bound. We propose a new reduction-based approach that enables us to obtain the first optimal rates (up to logarithmic factors) in the heavy-tailed setting, achieving error $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{n\epsilon})^{1 - \frac 1 k}$ under $(\epsilon, \delta)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}{\delta})$ factor, where $G_2^2$ and $G_k^k$ are the $2^{\text{nd}}$ and $k^{\text{th}}$ moment bounds on sample Lipschitz constants, nearly-matching a lower bound of [Lowy and Razaviyayn 2023]. We further give a suite of private algorithms in the heavy-tailed setting which improve upon our basic result under additional assumptions, including an optimal algorithm under a known-Lipschitz constant assumption, a near-linear time algorithm for smooth functions, and an optimal linear time algorithm for smooth generalized linear models.