@inproceedings{
asi2024private,
title={Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions},
author={Hilal Asi and Daogao Liu and Kevin Tian},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=oX6aIl9f0Y}
}
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.
Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions
Hilal Asi Daogao Liu Kevin Tian