Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers (2408.01336v1)

Abstract: We investigate a problem estimating coefficients of linear regression under sparsity assumption when covariates and noises are sampled from heavy tailed distributions. Additionally, we consider the situation where not only covariates and noises are sampled from heavy tailed distributions but also contaminated by outliers. Our estimators can be computed efficiently, and exhibit sharp error bounds.

• The paper introduces robust estimation methods combining thresholding and penalized Huber regression to address heavy-tailed covariates and noise.
• It demonstrates that the proposed estimators achieve convergence rates comparable to traditional lasso estimators under Gaussian settings.
• The research provides practical insights into scalable, tractable estimation techniques that advance high-dimensional sparse regression analysis.

Sparse Linear Regression in the Presence of Heavy-Tailed Distributions and Outliers

Overview

This essay provides an analytical overview of the paper titled "Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers", authored by Takeyuki Sasai and Hironori Fujisawa. The paper explores the challenging domain of estimating coefficients in sparse linear regression settings characterized by heavy-tailed distributions for both covariates and noise, compounded by the presence of outliers. The authors focus on constructing efficient estimators that achieve sharp error bounds within these complex settings.

Problem Statement

The intrinsic difficulties in sparse linear estimation are exacerbated when dealing with high-dimensional data featuring heavy-tailed distributions and contamination by outliers. Traditional methods often assume Gaussian or sub-Gaussian distributions for covariates and noise, presuming their robustness diminishes significantly in the presence of heavy-tailed distributions or outliers. Addressing this, the authors define two core models:

1. A standard sparse linear regression model accommodating heavy-tailed covariate and noise distributions.
2. A robust model allowing for adversarial outliers in the covariates and noise.

Methodology: Robust Estimation Techniques

The authors propose two algorithms for robust estimation:

1. ROBUST-SPARSE-ESTIMATION I: This algorithm addresses the scenario without outliers.
2. ROBUST-SPARSE-ESTIMATION II: This algorithm extends the first one to handle scenarios with outliers.

ROBUST-SPARSE-ESTIMATION I

The first algorithm mitigates the effect of heavy tails by employing a two-step procedure:

• THRESHOLDING: Caps the values of covariates to manage their tail behavior.
• PENALIZED-HUBER-REGRESSION: Combines the Huber loss function with L1 regularization to confer robustness against heavy tails and sparsity.

ROBUST-SPARSE-ESTIMATION II

This algorithm introduces additional preprocessing to mitigate the impact of outliers:

• COMPUTE-WEIGHT: Employs a semi-definite programming approach for sparse PCA to weigh data points, thereby reducing the influence of outliers.
• Follows THRESHOLDING and PENALIZED-HUBER-REGRESSION as in the first algorithm, but with preprocessed covariates.

Results: Theoretical Validation

The results presented in the paper include extensive theoretical derivations validating the proposed methods. Specifically, the authors derive non-asymptotic error bounds for the estimators under both heavy-tailed and contaminated conditions.

Theorem Highlights:

• Error Bounds for Standard Model: The authors prove that their estimator achieves a convergence rate similar to that of the lasso estimator under Gaussian assumptions but without requiring Gaussianity.
• Error Bounds for Robust Model: It is demonstrated that the estimator maintains robustness against outliers, with an error bound dependent on the tail properties and proportion of contamination.

Discussion

The estimators presented exhibit performance close to optimal in the presence of heavy-tailed distributions and outliers. A notable aspect of the results is the tractability of the estimators and the sharpness of the error bounds. Despite the rigorous theoretical grounding, achieving optimality in various dimensions, particularly in convergence rates and handling outliers, suggests several areas for future research:

• Independence from Norm Constraints: Current estimators' error bounds depend on norms of the true coefficients, which might be relaxed.
• Efficiency and Scalability: Future work could enhance the computational efficiency and make the estimator more scalable for very high-dimensional data.
• Optimal and Tractable Estimation: Developing estimators that are computationally efficient and theoretically optimal in handling heavy-tailed distributions and outliers remains a challenging and compelling direction.

Conclusion

The paper by Sasai and Fujisawa pushes the boundaries of sparse linear regression by systematically addressing the combined challenges of heavy-tailed distributions and outliers. Their methodologically robust approach, supported by strong theoretical guarantees, provides a significant step toward more resilient and effective high-dimensional estimators. Future work in this area, particularly focusing on improved computational methods and theoretical efficiencies, holds promise for further advancements in statistical learning methodologies under challenging conditions.