Adversarial Contamination Meets Hard Thresholding: An Iterative Algorithm with Signal Adaptivity and Minimax Optimality

Pervasive data contamination -- stemming from measurement errors, outliers, or adversarial corruption -- has motivated the development of robust statistical methods.

In this context, we propose a two-stage Adversarial Contamination-resistant Iterative Hard Thresholding (AC-IHT) algorithm for high-dimensional regression with contamination.

Our nonconvex algorithm achieves minimax near-optimal (up to logarithmic terms) estimation by iteratively updating the coefficient vector and the contamination vector with different thresholding scales.

We further demonstrate that our AC-IHT estimator is signal-adaptive: under proper signal conditions, it adaptively attains a sharper estimation rate and more accurate support recovery.

Moreover, it enjoys the strong oracle property, laying a theoretical foundation for asymptotic inference.

Numerical experiments confirm its superior finite-sample performance.

Finally, we discuss theoretical extensions of the proposed procedure to generalized linear models and to heavy-tailed noise settings.