Adaptive Robust Confidence Intervals

2024-10-30

AI Summary

This paper addresses the problem of constructing adaptive robust confidence intervals for the Gaussian mean under the Huber contamination model with unknown contamination proportion. It establishes, for the first time, that any adaptive interval must suffer an exponential degradation in length compared to the non-adaptive optimal interval achievable when the contamination level is known—and crucially, this fundamental lower bound depends intrinsically on the shape of the underlying data distribution. To overcome this challenge, the authors propose a novel construction based on joint uncertainty quantification over all contamination levels via full-level quantiles. They develop a complete optimal adaptive theory for the Gaussian location model and a broad class of generalized robust tests, deriving tight information-theoretic lower bounds and providing a computationally feasible, rate-optimal procedure. The core contribution lies in characterizing the distribution-dependent price of adaptivity and delivering the first framework that simultaneously achieves theoretical optimality and practical implementability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper studies the construction of adaptive confidence intervals under Huber's contamination model when the contamination proportion is unknown. For the robust confidence interval of a Gaussian mean, we show that the optimal length of an adaptive interval must be exponentially wider than that of a non-adaptive one. An optimal construction is achieved through simultaneous uncertainty quantification of quantiles at all levels. The results are further extended beyond the Gaussian location model by addressing a general family of robust hypothesis testing. In contrast to adaptive robust estimation, our findings reveal that the optimal length of an adaptive robust confidence interval critically depends on the distribution's shape.

Problem

• Construct adaptive confidence intervals under unknown contamination proportion
• Determine optimal length for robust Gaussian mean confidence intervals
• Extend results to general robust hypothesis testing beyond Gaussian models

Innovation

• Adaptive confidence intervals under contamination
• Simultaneous uncertainty quantification of quantiles
• Optimal length depends on distribution shape