Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions
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| Format: | Preprint |
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2024
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| author | Asi, Hilal Liu, Daogao Tian, Kevin |
| author_facet | Asi, Hilal Liu, Daogao Tian, Kevin |
| contents | 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 $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{nε})^{1 - \frac 1 k}$ under $(ε, δ)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}δ)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_02789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions Asi, Hilal Liu, Daogao Tian, Kevin Data Structures and Algorithms Cryptography and Security Machine Learning 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 $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{nε})^{1 - \frac 1 k}$ under $(ε, δ)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}δ)$ 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. |
| title | Private Stochastic Convex Optimization with Heavy Tails: Near-Optimality from Simple Reductions |
| topic | Data Structures and Algorithms Cryptography and Security Machine Learning |
| url | https://arxiv.org/abs/2406.02789 |