Adaptive Robust Confidence Sets in High Dimensions

Abstract:We study adaptive confidence sets for mean vectors of multivariate Gaussians with identity covariance under Huber contamination when the contamination proportion is unknown. We show that an information-theoretically optimal adaptive confidence set can be constructed by simultaneously quantifying uncertainty across all one-dimensional projections. The resulting confidence set achieves the optimal diameter and reveals a substantial adaptation cost compared with the minimax estimation error. Although statistically optimal, this construction is computationally infeasible. We establish a Statistical Query (SQ) lower bound showing that achieving diameter $O(m^{-1/2})$ requires either exponentially many queries or query tolerance $d^{-\Omega(m)}$. Thus, in contrast to robust mean estimation, adaptation to the unknown contamination proportion incurs both a statistical cost and a computational barrier within the SQ framework. Finally, we give a low-degree moment-matching algorithm that constructs an adaptive confidence set using $d^{O(m)}$ samples and time. Its diameter bound contains an additional $O(m^{-1/2})$ term beyond the information-theoretic rate, matching the dependence on $m$ in the SQ lower bound. Together, these results characterize the statistical limits of adaptive robust confidence sets under Huber contamination and quantify a computational barrier within the SQ framework.

Statistics Theory (math.ST); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Methodology (stat.ME); Machine Learning (stat.ML)

arXiv:2610.15723 [math.ST]

(or arXiv:2610.15723v1 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2610.15723 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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