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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.06492 |
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| _version_ | 1866913090254143488 |
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| author | Lowy, Andrew |
| author_facet | Lowy, Andrew |
| contents | We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure $\varepsilon$-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded $k$-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. Prior work characterized the minimax optimal rate for $ρ$-zero-concentrated DP SCO up to logarithmic factors in this setting, but the pure $\varepsilon$-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure $\varepsilon$-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in deterministic polynomial time when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes -- including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and polytopes -- we achieve deterministic polynomial time even when the worst-case Lipschitz parameter is infinite. Our approach is based on a novel framework for privately optimizing Lipschitz extensions of the empirical loss. We complement our upper bound with a nearly matching high-probability lower bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06492 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Rates for Pure $\varepsilon$-Differentially Private Stochastic Convex Optimization with Heavy Tails Lowy, Andrew Machine Learning Cryptography and Security We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure $\varepsilon$-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded $k$-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. Prior work characterized the minimax optimal rate for $ρ$-zero-concentrated DP SCO up to logarithmic factors in this setting, but the pure $\varepsilon$-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure $\varepsilon$-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in deterministic polynomial time when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes -- including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and polytopes -- we achieve deterministic polynomial time even when the worst-case Lipschitz parameter is infinite. Our approach is based on a novel framework for privately optimizing Lipschitz extensions of the empirical loss. We complement our upper bound with a nearly matching high-probability lower bound. |
| title | Optimal Rates for Pure $\varepsilon$-Differentially Private Stochastic Convex Optimization with Heavy Tails |
| topic | Machine Learning Cryptography and Security |
| url | https://arxiv.org/abs/2604.06492 |