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An integrated perspective of robustness in regression through the lens of the bias-variance trade-off

Akifumi Okuno

TL;DR

This work reframes regression robustness by linking outlier-resistant estimation (optimistic regression) and robust optimization (pessimistic regression) through a bias-variance lens. It introduces an integrated estimator parameterized by a scalar lambda that blends optimistic and pessimistic losses and analyzes its generalization behavior via a bias-variance decomposition. Through simulations, it shows that small optimism can sharply reduce variance in the presence of outliers while excessive optimism or pessimism can inflate bias or variance, depending on model specification. The study highlights practical guidance: use modest pessimism or balanced strategies when misspecification is possible, and provides reproducible code for further exploration of the bias-variance trade-off in robust regression.

Abstract

This paper presents an integrated perspective on robustness in regression. Specifically, we examine the relationship between traditional outlier-resistant robust estimation and robust optimization, which focuses on parameter estimation resistant to imaginary dataset-perturbations. While both are commonly regarded as robust methods, these concepts demonstrate a bias-variance trade-off, indicating that they follow roughly converse strategies.

An integrated perspective of robustness in regression through the lens of the bias-variance trade-off

TL;DR

This work reframes regression robustness by linking outlier-resistant estimation (optimistic regression) and robust optimization (pessimistic regression) through a bias-variance lens. It introduces an integrated estimator parameterized by a scalar lambda that blends optimistic and pessimistic losses and analyzes its generalization behavior via a bias-variance decomposition. Through simulations, it shows that small optimism can sharply reduce variance in the presence of outliers while excessive optimism or pessimism can inflate bias or variance, depending on model specification. The study highlights practical guidance: use modest pessimism or balanced strategies when misspecification is possible, and provides reproducible code for further exploration of the bias-variance trade-off in robust regression.

Abstract

This paper presents an integrated perspective on robustness in regression. Specifically, we examine the relationship between traditional outlier-resistant robust estimation and robust optimization, which focuses on parameter estimation resistant to imaginary dataset-perturbations. While both are commonly regarded as robust methods, these concepts demonstrate a bias-variance trade-off, indicating that they follow roughly converse strategies.
Paper Structure (12 sections, 2 theorems, 20 equations, 10 figures)

This paper contains 12 sections, 2 theorems, 20 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Symbols and Notation
  4. Optimistic Regression and Pessimistic Regression
  5. OLS regression
  6. Optimistic regression
  7. Pessimistic Regression
  8. Integration of Optimistic Regression and Pessimistic Regression
  9. Bias and variance terms in generalization error
  10. Bias-variance trade-off between optimistic and pessimistic linear regression
  11. Further discussion
  12. Conclusion

Key Result

Proposition 1

It holds for the Huber's function $\rho_{\eta}(u)=(u^2/2) \mathbbm{1}_{|u| \le \eta}+\{\eta|u|-\eta^2/2\}\mathbbm{1}_{|u|>\eta}$ that $\mathsf{Y}^{\min}_{1,1,\tau}(\boldsymbol y,\boldsymbol{f_{\boldsymbol \theta}}(\boldsymbol X)) \propto \sum_{i=1}^{n}\rho_{n/2d\tau}(y_i-f_{\boldsymbol \theta}(\bold

Figures (10)

  • Figure 1: Bias-variance trade-off between optimistic regression (i.e., outlier-robust estimation, where increasing $\lambda>0$ enhances the optimism) and pessimistic regression (i.e., robust optimization, where decreasing $\lambda<0$ enhances the pessimism). $\lambda=0$ represents OLS. This figure is a copy of Figure \ref{['fig:exp1_default']}(\ref{['subfig:default_(c)']}); see Section \ref{['sec:experiments']} for details.
  • Figure 2: Covariate noise
  • Figure 3: Outcome outliers
  • Figure 4: None
  • Figure 6: Covariate noise
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 1: A nonlinear extension of gannaz2007robust
  • Proposition 2: ribeiro2023regularization Proposition 1