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We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise — where an ε-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ε) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of and the running time is polynomial in d and 1/ε. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ε. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.
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This content will become publicly available on January 1, 2026
Optimal Rates for Robust Stochastic Convex Optimization
Machine learning algorithms in high-dimensional settings are highly susceptible to the influence of even a small fraction of structured outliers, making robust optimization techniques essential. In particular, within the ε-contamination model, where an adversary can inspect and replace up to an ε-fraction of the samples, a fundamental open problem is determining the optimal rates for robust stochastic convex optimization (SCO) under such contamination. We develop novel algorithms that achieve minimax-optimal excess risk (up to logarithmic factors) under the ε-contamination model. Our approach improves over existing algorithms, which are not only suboptimal but also require stringent assumptions, including Lipschitz continuity and smoothness of individual sample functions. By contrast, our optimal algorithms do not require these stringent assumptions, assuming only population-level smoothness of the loss. Moreover, our algorithms can be adapted to handle the case in which the covariance parameter is unknown, and can be extended to nonsmooth population risks via convolutional smoothing. We complement our algorithmic developments with a tight information-theoretic lower bound for robust SCO.
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- Award ID(s):
- 2224213
- PAR ID:
- 10628396
- Editor(s):
- Bun, Mark
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 329
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-367-6
- Page Range / eLocation ID:
- 9:1-9:21
- Subject(s) / Keyword(s):
- Adversarial Robustness Machine Learning Optimization Algorithms Robust Optimization Stochastic Convex Optimization Theory of computation → Mathematical optimization
- Format(s):
- Medium: X Size: 21 pages; 881891 bytes Other: application/pdf
- Size(s):
- 21 pages 881891 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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