Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions
Keywords: Stochastic Convex Optimization, Heavy-Tailed Distributions, Differential Privacy
TL;DR: We give the first algorithm attaining near-optimal error rates for DP-SCO assuming heavy-tailed gradients, and several improvements in structured cases.
Abstract: 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.
Primary Area: Privacy
Submission Number: 18826
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