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Adversarial robust weighted Huber regression

Takeyuki Sasai, Hironori Fujisawa

Abstract

We consider a robust estimation of linear regression coefficients. In this note, we focus on the case where the covariates are sampled from an $L$-subGaussian distribution with unknown covariance, the noises are sampled from a distribution with a bounded absolute moment and both covariates and noises may be contaminated by an adversary. We derive an estimation error bound, which depends on the stable rank and the condition number of the covariance matrix of covariates with a polynomial computational complexity of estimation.

Adversarial robust weighted Huber regression

Abstract

We consider a robust estimation of linear regression coefficients. In this note, we focus on the case where the covariates are sampled from an -subGaussian distribution with unknown covariance, the noises are sampled from a distribution with a bounded absolute moment and both covariates and noises may be contaminated by an adversary. We derive an estimation error bound, which depends on the stable rank and the condition number of the covariance matrix of covariates with a polynomial computational complexity of estimation.

Paper Structure

This paper contains 16 sections, 10 theorems, 79 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Method
  3. COMPUTE-WEIGHT
  4. THRESHOLDING
  5. WEIGHTED-HUBER-REGRESSION
  6. Result
  7. Key propositions
  8. Proofs of the main theorems
  9. Proofs of Proposition \ref{['t:main']}-\ref{['p:main:sc']}
  10. Proof of Proposition \ref{['t:main']}
  11. Proofs of Propositions \ref{['p:main1']}, \ref{['p:main:out']}, \ref{['p:main:out2']} and \ref{['p:main:sc']}
  12. Some tools
  13. Proof of proposition \ref{['p:main1']}
  14. Proof of Proposition \ref{['p:main:out']}
  15. Proof of Proposition \ref{['p:main:out2']}
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

Suppose that $\left\{\mathbf{x}_i \right\}_{i=1}^n$ is a sequence with i.i.d. random vectors drawn from an $L$-subGaussian distribution with mean zero and with covariance matrix $\Sigma$, where $\Sigma$ is unknown. Suppose that $\left\{\xi_i\right\}_{i=1}^n$ is a sequence with i.i.d. random variable where $c(L,\sigma)$ is some constant depending on $L$ and $\sigma$ and $r_\Sigma = \mathrm{Trace}(\

Theorems & Definitions (17)

  • Theorem \oldthetheorem
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Definition 3.1: $L$-subGaussian random vector
  • Definition 3.2: $\psi_2$-norm
  • Remark 3.1
  • Theorem \oldthetheorem
  • Proposition 4.1
  • Proposition 4.2
  • ...and 7 more