Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions

• URL: http://arxiv.org/abs/2406.02789v1
• Date: Tue, 4 Jun 2024 21:26:29 GMT
• Title: Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions
• Authors: Hilal Asi, Daogao Liu, Kevin Tian,
• Abstract summary: We study the problem of differentially private convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a $ktextth$-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 $Gcdot frac 1 sqrt n + G_k cdot (fracsqrt dnepsilon)1
• Score: 19.008521454738425
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 [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.

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