High Dimensional Robust M-Estimation: Asymptotic Variance via Approximate Message Passing (1310.7320v3)

Abstract: In a recent article (Proc. Natl. Acad. Sci., 110(36), 14557-14562), El Karoui et al. study the distribution of robust regression estimators in the regime in which the number of parameters p is of the same order as the number of samples n. Using numerical simulations and highly plausible' heuristic arguments, they unveil a striking new phenomenon. Namely, the regression coefficients contain an extra Gaussian noise component that is not explained by classical concepts such as the Fisher information matrix. We show here that that this phenomenon can be characterized rigorously techniques that were developed by the authors to analyze the Lasso estimator under high-dimensional asymptotics. We introduce an approximate message passing (AMP) algorithm to compute M-estimators and deploy state evolution to evaluate the operating characteristics of AMP and so also M-estimates. Our analysis clarifies that theextra Gaussian noise' encountered in this problem is fundamentally similar to phenomena already studied for regularized least squares in the setting n<p.

• The paper characterizes a novel extra Gaussian noise component affecting the asymptotic variance of M-estimators in high-dimensional settings.
• It proposes an Approximate Message Passing (AMP) algorithm tailored for M-estimation, which reveals this extra noise component as a fixed point.
• The findings highlight that classical estimators are statistically inefficient in high dimensions and necessitate new algorithmic approaches like AMP for accurate inference.

High Dimensional Robust M-Estimation: Asymptotic Variance via Approximate Message Passing

The paper "High Dimensional Robust M-Estimation: Asymptotic Variance via Approximate Message Passing" by David Donoho and Andrea Montanari explores the statistical properties of M-estimators in high-dimensional settings, where the number of parameters $p$ approaches the number of data points $n$. The authors rigorously demonstrate the existence of an extra Gaussian noise component affecting the asymptotic variance of regression coefficients, a phenomenon previously noted by El Karoui et al. using heuristic arguments. The authors leverage techniques used to analyze the Lasso estimator under high-dimensional conditions, developing an Approximate Message Passing (AMP) algorithm that elucidates this noise.

Key Contributions

1. Characterization of Extra Gaussian Noise: Traditional theory on M-estimators assumes $p$ is fixed while $n$ increases, with asymptotic behavior encapsulated in classical variance formulas. In the high-dimensional scenario where $n$ and $p$ grow simultaneously, Donoho and Montanari identify a novel noise component, $\tau_*^2$, characterized by a convolution of the original noise with a Gaussian distribution.
2. Approximate Message Passing for M-Estimation: The authors propose an AMP algorithm tailored for M-estimation, yielding the M-estimator as a fixed point. This algorithm iteratively refines estimates by considering effective score functions that incorporate the additional noise component, thus allowing for a precise evaluation of the estimator properties.
3. State Evolution Analysis: The work extends the notion of state evolution, a tool for tracking algorithmic behavior across iterations in high-dimensional problems, enabling them to predict the performance of the AMP algorithm rigorously.
4. Implications on Statistical Efficiency: The persistence of extra Gaussian noise underlines inefficiencies in classical estimators when applied to high-dimensional datasets, often leading to erroneous confidence assessments.

Numerical Results

The AMP algorithm's efficacy is shown using synthetic datasets, where the high-dimensional behavior is observed to deviate from classical expectations. The effective noise's impact on estimator variance is quantified, and AMP's ability to capture this is validated by empirical results aligning closely with theoretical predictions.

Theoretical Implications

The work holds significant implications for theoretical statistics, especially on the limits of traditional estimation techniques in the high-dimensional regime. The authors highlight the necessity for revised statistical tools when handling data-rich environments, ensuring that high-dimensional expectations are accurately integrated into inferential procedures.

Practical Implications

For practitioners handling high-dimensional data, the paper's insights stress the importance of accounting for the increased covariance due to the extra Gaussian component. Estimation strategies that overlook this may lead to underpowered inference and inefficiencies in applications ranging from bioinformatics to signal processing. This necessitates practical deployment of algorithms like AMP, designed to robustly navigate the intricacies of high-dimensional data.

Conclusion and Future Work

This paper successfully bridges the gap between theoretical understanding and practical application in high-dimensional statistics. Future research could explore extensions of this analysis to non-Gaussian designs, incorporating heterogeneous noise scenarios and broadening beyond M-estimators to include penalized variants. Additionally, potential extensions may involve adapting the AMP framework for real-world applications across diverse domains, further enriching the methodological toolkit available for data scientists and statisticians.